How Levins’ dynamics emerges from a Ricker metapopulation model

Abstract

Understanding the dynamics of metapopulations close to extinction is of vital importance for management. Levins-like models, in which local patches are treated as either occupied or empty, have been used extensively to explore the extinction dynamics of metapopulations, but they ignore the important role of local population dynamics. In this paper, we consider a stochastic metapopulation model where local populations follow a stochastic, density-dependent dynamics (the Ricker model), and use this framework to investigate the behaviour of the metapopulation on the brink of extinction. We determine under which circumstances the metapopulation follows a time evolution consistent with Levins’ dynamics. We derive analytical expressions for the colonisation and extinction rates (c and e) in Levins-type models in terms of reproduction, survival and dispersal parameters of the local populations, providing an avenue to parameterising Levins-like models from the type of information on local demography that is available for a number of species. To facilitate applying our results, we provide a numerical algorithm for computing c and e.

Appendices

Appendix A

In this appendix, we derive an expression for P _j→i in Eq. 4, the probability that a patch with j inhabitants in generation t has i inhabitants in generation t+1. We calculate P _j→i as follows. First, consider reproduction. Take a patch with j adult inhabitants. As described in the main text, each individual independently gives rise to a Poisson distributed number n of offspring with mean R. Conditional on the number of adults, j, the number n of offspring in this patch is thus Poisson distributed with mean R j, $\mathcal {P}(n|j) =\text {Poisson}(n, Rj)$. Each offspring independently survives cannibalism by the adults with probability exp(−α j). Conditional on the number of offspring, n say, the number k of surviving offspring is binomially distributed

$$ \mathcal{P}(k| n,j) = \left( \begin{array}{c}n\\k \end{array}\right) (\mathrm{e}^{-\alpha j})^{k} (1-\mathrm{e}^{-\alpha j})^{n-k}. $$

(A.1)

Since $\mathcal {P}(k|j) = {\sum }_{n} \mathcal {P}(n|j) \mathcal {P}(k|n,j) = \text {Poisson}$ (k,R j exp(−α j)), we conclude that the number of offspring surviving cannibalism is Poisson distributed, Poisson(k,λ) with parameter λ=R j exp(−α j). Now consider dispersal. Offspring emigrate independently with probability m. For the patch in question, denote the number of emigrating offspring by k _e and the number of resident offspring by k _r. We have that k _e+k _r=k. Since k is Poisson distributed, it follows that k _e and k _r are independently Poisson distributed (conditional on a given number j of adults):

$$\begin{array}{@{}rcl@{}} \mathcal{P}(k_{\mathrm{e}},k_{\mathrm{r}}) &=& \sum\limits_{k} \text{Poisson}(k,\lambda) \left( \begin{array}{c}k\\k_{\mathrm{e}} \end{array}\right) m^{k_{\mathrm{e}}} (1-m)^{k_{\mathrm{r}}} \delta_{k,k_{\mathrm{e}}+k_{\mathrm{r}}}\\ & =&\text{Poisson}(k_{\mathrm{e}},m\lambda)\text{Poisson}(k_{\mathrm{r}},(1-m)\lambda). \end{array} $$

(A.2)

Now consider immigration. As explained in the main text, the emigrants from all patches are gathered in a common pool from where the target patch of each emigrant is independently and randomly chosen, and thus the emigrants are uniformly randomly distributed among all N patches. It follows from the independence of k _e and k _r for a given patch that the numbers of resident and immigrant offspring are independently distributed, conditional on the frequencies f _j . We remark that in the limit of infinitely many patches considered here, the frequency f _j is equal to the probability that a randomly chosen patch contains j individuals. Since dispersing individuals choose their destination independently, the number of offspring dispersing into a given patch is Poisson-distributed with expected value

$$ D(t) = m \sum\limits_{j=1}^{\infty} R j \exp(-\alpha j) f_{j}(t)\,. $$

(A.3)

Bringing it all together, the probability P _j→i of finding i individuals in the patch in the next generation starting from j individuals in the current generation is found to be:

$$\begin{array}{@{}rcl@{}} P_{j \to i} = \frac{\left[(1-m) Rj \mathrm{e}^{-\alpha j} + D(t)\right]^{i}}{i!}\exp\left\{ -\left[(1-m) Rj \mathrm{e}^{-\alpha j} +D(t)\right]\right\}, \quad \text{for }i,j=0,1,2,\dotsc~. \end{array} $$

(A.4)

Note that P _j→i depends upon f _j (t) through the number of immigrants, D(t).

Appendix B

In this section, we describe the procedure to calculate the critical migration rate m _c, below which the metapopulation becomes deterministically extinct. In order to determine how m _c depends upon the life history parameters R and α, we investigate the linear stability of the extinction steady state f=0 (which is a steady state for any given set of parameters since G _i (f=0)=0 for i≥1). We denote the linear stability matrix by A. Its elements are

$$ A_{ij} = \frac{\partial G_{i}}{\partial f_{j}} \, . $$

(B.1)

To determine the linear stability of the state f=0, we must evaluate A at f=0. The corresponding stability matrix A ⁽⁰⁾ has the elements

$$ A_{1j}^{(0)} = \rho_{j} \mathrm{e}^{-\rho_{j}} + m_{\mathrm{c}} {\Lambda}_{j} \quad\text{and}\quad A_{ij}^{(0)} = \frac{{\rho_{j}^{i}}}{i!} \mathrm{e}^{-\rho_{j}} \quad \text{for }i > 1, $$

(B.2)

where we introduced Λ _j =R je^−αj and ρ _j =(1−m _c)Λ _j to simplify the notation of Eq. B.2. The state f=0 is linearly stable provided all the eigenvalues λ _j of A ⁽⁰⁾ are less than unity in modulus. This state turns unstable when the leading eigenvalue (the largest in absolute value) approaches unity: λ ₁(m)→1 as m→m _c. This criterion allows us to compute m _c in the following manner. We truncate the matrix A ⁽⁰⁾ to linear size j _max, where j _max is chosen such that ${\sum }_{i,j=j_{\max }}^{\infty }A_{ij}^{(0)}\approx 0$, consistent with the truncation of Eq. 4 specified in the main text. We then numerically compute the eigenvalues of the truncated matrix for a suitable range of values of m, thus constructing a function λ ₁(m). Finally, from this function, we interpolate the value of m=m _c such that λ ₁(m _c)=1.

Appendix C

In this section, we show how Levins’ dynamics (9, 10) emerges from the deterministic dynamics (4). The dynamics (4) simplifies for small positive values of the parameter δ. This parameter is defined in Eq. 7 as the relative difference of the dispersal rate m from the brink of extinction m _c.

In order to show how the dynamics simplifies on the brink of extinction, we follow Dykman et al. (1994) and Eriksson et al. (2013) and expand the right-hand side of Eq. 4 in powers of δ. We assume that the deviations f _i (t) from 0 remain small. It turns out that they are in fact of order δ. This will become clear at the end of this appendix. To simplify the notation, we define Δf _i (t+1)=f _i (t+1)−f _i (t) and drop the generation dependence t. We find:

$$\begin{array}{@{}rcl@{}} {\Delta} f_{i} &\approx& - f_{i} + \sum\limits_{j} A_{ij}^{(0)} f_{j} + \delta \sum\limits_{j} A_{ij}^{(1)} f_{j} + \frac{1}{2} \sum\limits_{jk} A_{ijk}^{(2)} f_{j} f_{k}\\ &&\qquad + \frac{1}{2}\delta \sum\limits_{jk} A_{ijk}^{(3)} f_{j} f_{k} \cdots \end{array} $$

(C.1)

The coefficients ${A}^{(1)}_{ij}$ are given by

$$\begin{array}{@{}rcl@{}} A_{1j}^{(1)} &= & m_{\mathrm{c}} {\Lambda}_{j} \mathrm{e}^{-\rho_{j} } \left( \rho_{j} - 1 \right) + m_{\mathrm{c}}{\Lambda}_{j} , \\ A_{ij}^{(1)} &= & m_{\mathrm{c}} {\Lambda}_{j} \mathrm{e}^{-\rho_{j} } \sum\limits_{k=i-1}^{i} \frac{{\rho_{j}}^{k}}{k!}(-1)^{i-k} \quad\text{for }i > 1, {c2} \end{array} $$

(C.2)

where Λ _j =R je^−αj and ρ _j =(1−m _c)Λ _j as in Appendix B. The coefficients ${A}_{ijk}^{(2)}$ read:

$$\begin{array}{@{}rcl@{}} A_{1jk}^{(2)}& = & m_{\mathrm{c}} {\Lambda}_{k} \mathrm{e}^{- \rho_{j} } \left( 1 - \rho_{j} \right) + m_{\mathrm{c}} {\Lambda}_{j} \mathrm{e}^{- \rho_{k} } \left( 1 - \rho_{k} \right)\\ &&\qquad- m_{\mathrm{c}} \left( {\Lambda}_{j} + {\Lambda}_{k} + 2 m_{\mathrm{c}} {\Lambda}_{j} {\Lambda}_{k}\right), \\ A_{2jk}^{(2)} &= & m_{\mathrm{c}} {\Lambda}_{k}\mathrm{e}^{- \rho_{j} } \left( \rho_{j} - \frac{{\rho_{j}}^{2}}{2} \right) + m_{\mathrm{c}} {\Lambda}_{j} \mathrm{e}^{- \rho_{k} } \left( \rho_{k} \!- \!\frac{{\rho_{k}}^{2}}{2} \right)\\ &&\qquad+ m_{\mathrm{c}}^{2} {\Lambda}_{j} {\Lambda}_{k}, \\ A_{ijk}^{(2)} &= & m_{\mathrm{c}} {\Lambda}_{k} \mathrm{e}^{- \rho_{j} } \sum\limits_{\nu=i-1}^{i} \frac{ {\rho_{j}}^{\nu}}{\nu!}(-1)^{\nu-i+1}\\ &&+ m_{\mathrm{c}} {\Lambda}_{j} \mathrm{e}^{- \rho_{k} } \sum\limits_{\nu=i-1}^{i} \frac{{\rho_{k}}^{\nu}}{\nu!}(-1)^{\nu-i+1} \text{for }i > 2. \end{array} $$

(C.3)

The last term in the expansion (C.1) is of order δ ³. There are a number of terms of the same order, δ ³, that are neglected in Eq. C.1. These terms are negligible because they are of second or higher order in m _c. This critical migration rate is exponentially small in the limit where the local dynamics is fast and dispersal is slow as Fig. 1 shows. In this limit, $m_{\mathrm {c}}^{2}$-terms can be neglected in Eq. C.1. Additionally, it turns out that the coefficients $A^{(3)}_{ijk}$ are given by

$$ A^{(3)}_{ijk} = A^{(2)}_{ijk} + O(m_{\mathrm{c}}^{2}). $$

(C.4)

Thus, Eq. C.1 simplifies to

$$ {\Delta} f_{i} \approx - f_{i} + \sum\limits_{j} A_{ij}^{(0)} f_{j} + \delta \sum\limits_{j} A_{ij}^{(1)} f_{j} + \frac{1}{2}(1+\delta) \sum\limits_{jk} A_{ijk}^{(2)} f_{j} f_{k} + \cdots $$

(C.5)

The δ ³-term that is kept in Eq. C.5 contains terms linear in m _c as Eq. C.3 shows.

The metapopulation dynamics becomes essentially one-dimensional near the brink of extinction (Eriksson et al. 2013). One particular linear combination of the f _j changes slowly, and all others relax quickly to equilibrium positions that depend on the slow variable. In order to identify the slow variable, we diagonalise A ⁽⁰⁾:

$$ \mathbf{A}^{(0)} \mathbf{R}_{\beta} = \lambda_{\beta} \mathbf{R}_{\beta},\quad \mathbf{L}_{\beta}^{\mathsf{T}} \mathbf{A}^{(0)} = \mathbf{L}_{\beta}^{\mathsf{T}}\lambda_{\beta},\quad \text{for }\beta = 1,2,\dotsc. $$

(C.6)

The eigenvalues λ _β are ordered as |λ ₁|>|λ ₂|>|λ ₃|>⋯. We take the left and right eigenvectors to be bi-orthonormal

$$ \mathbf{L}_{\beta}^{\mathsf{T}} \mathbf{R}_{\gamma}= \delta_{\beta\gamma}~. $$

(C.7)

The condition (C.7) does not determine the normalisation of the eigenvectors. Our normalisation condition is discussed below.

At δ=0, we have that λ ₁=1. For small positive values of δ, we find that λ ₁ is real and slightly larger than unity, λ ₁=1+O(δ), and |λ _β |<1 for β>1. Thus, for small values of δ, perturbations away from f=0 grow slowly in the direction of L ₁, and decay rapidly in the directions L _β , β>1. Thus the slow variable is given by

$$ Q_{1} = \mathbf{L}_{1}^{\mathsf{ T}}\boldsymbol{f}, $$

(C.8)

and the fast variables are given by

$$ Q_{\beta} = \mathbf{L}_{\beta}^{\mathsf{ T}} \boldsymbol{f} \quad \text{for }\beta > 1. $$

(C.9)

The fast variables quickly approach local equilibria that depend on the instantaneous value of the slow variable, Q ₁. These local equilibria take small values of order O(δ ²). This can be shown by inserting (C.9) into (C.5) and solving for ΔQ _β =0, β>1. It thus follows from Eqs. C.7, C.8 and C.9 that

$$ f_{j}(t) = \sum\limits_{\beta=1}^{\infty}R_{\beta j}Q_{\beta}(t) \approx R_{1j}Q_{1}(t) + O(\delta^{2})~. $$

(C.10)

It remains to derive an equation for the dynamics of Q ₁. Inserting (C.10) into Eq. C.5, we find

$$ {\Delta} Q_{1} = {a_{1}}\delta ~Q_{1} + {a_{2}}(1 + \delta){Q_{1}^{2}}~, $$

(C.11)

where the coefficients a ₁ and a ₂ are given by

$$\begin{array}{@{}rcl@{}} a_{1}&=& \sum\limits_{ij} L_{1i} A_{ij}^{(1)} R_{1j} , \end{array} $$

(C.12)

$$\begin{array}{@{}rcl@{}} a_{2}&=&\frac{1}{2} \sum\limits_{ijk} L_{1i} A_{ijk}^{(2)} R_{1j} R_{1k} . \end{array} $$

(C.13)

Equation C.11 has the form of Levins’ equation (for discrete time evolution):

$$ {\Delta} Q_{1}= cQ_{1}\left( 1-Q_{1}\right) - e Q_{1}~ . $$

(C.14)

The colonisation and extinction rates are given by:

$$ c = -{a_{2}}(1+\delta) ~ \quad \text{and} \quad e = c - {a_{1}}\delta~ . $$

(C.15)

When the local population dynamics is given by a time-continuous birth-death process, the components L _1j of L ₁ approach a j-independent constant for large values of j (L _1j →const. as j→∞), as shown in Eriksson et al. (2013). We expect the same to be true in the present model. This is borne out by numerical evidence shown in Fig. 6a. More examples are given in Fig. ?? in the supplementary material. We normalise the leading left eigenvector L ₁ by taking this constant to be unity. We remark that in this normalisation $\mathbf {L}_{1}^{\mathsf { T}} \rightarrow (1,1,1,\dotsc )$ as R becomes large and α tends to zero. Thus, in this limit the slow variable corresponds to the fraction of occupied patches. Figure 6a shows that L _1j ≈1 for j>15 when R=1.25 and α=0.007. Finally, given L ₁, the normalisation of R ₁ is determined by the condition (C.7). Panel b shows the components of R ₁ corresponding to L ₁ displayed in panel a. We note that a ₁ is positive while a ₂ is negative. Moreover, as R becomes large and α→0, the coefficient a ₁ is approximately equal to −a ₂ (see Fig. ?? in the supplementary material). This is also expected, as it is the case in the time-continuous process studied in Eriksson et al. (2013). This implies that the coefficient e is approximately independent of δ, while c scales as 1+δ, see Fig. 5a in the main text. This result can be interpreted as follows. On the brink of extinction, with the local patch dynamics being much faster than the migration dynamics, a slight increase in migration will directly increase the rate of colonisation events, as more individuals arrive to empty patches. The number of extinction events will, however, remain the same, since the number of expected immigrants to a given occupied patch is too small to increase its expected time to extinction.

Fig. 6

Components L _1j of leading left eigenvector L ₁ as a function of j, for R=1.25. The values of α are given in the figure (same as in Fig. 1). b Components R _1j of leading right eigenvector R ₁ as a function of j, same parameter values as in panel (a)

Equation C.8 shows that the slow variable Q ₁(t) is a linear combination of the frequencies f _j (t). This equation implies that the slow variable corresponds to the fraction of occupied patches if $\mathbf {L}_{1}^{\mathsf { T}} = (1,1,1,\dotsc )$. The normalisation convention discussed above ensures that Q ₁ can be interpreted as the fraction of occupied patches in the limit of large R and small α.

The relation (C.10) above determines f _j (t) in terms of Q ₁(t), and can be used to reconstruct the dynamics of f. It shows that the coefficients w _j in Eq. 8 are approximately given by the components R _1j of the right eigenvector R ₁ corresponding to the leading eigenvalue λ ₁=1. Comparing (C.8) with Eq. 8 in the main text, we see that

$$ w_{j} = R_{1j}~. $$

(C.16)

Finally, we note that Eq. C.10 shows that the components f _j are indeed of order δ, as assumed in the beginning of this Section.

A numerical implementation of our method for computing c and e is available as a MATLAB script in the supplementary material. The fixed points of Eq. C.14 are the extinction point Q ₁ = 0 and the steady-state $Q_{1}^{\ast } = 1- e/c$. For |c−e|<2 and c<e, the extinction point 0 is stable and the steady state $Q_{1}^{\ast }$ is unstable. At c=e, both fixed points meet and exchange their stability for c>e. At |c−e|=2 a period-doubling bifurcation occurs (Ott 2002), and as |c−e| increases still further, chaos ensues. See Hastings and Higgins (1994) for the study of chaotic transients. However, this behaviour is outside the range of applicability of Eq. C.14, that is, too far from the critical line in Fig. 1.

We conclude by summarising the three assumptions that were necessary to obtain (C.14) and (C.15). First, it was assumed that the number N of patches is large. Second, it was assumed that the metapopulation is on the brink of extinction (allowing us to assume that the parameter δ is small). Third, it was assumed that the local dynamics is much faster than the global dynamics (allowing us to assume that m _c is small).