Dispersal traits interact with dynamic connectivity to affect metapopulation growth and stability

Abstract

Many marine benthic species undergo a pelagic larval stage during which larvae are transported by ocean currents over a broad range of spatial and temporal scales. Although metapopulation theory predicts how stochastic dispersal can alter the stability of metapopulations, little is known about how dispersal-related traits such as spawning time and larval duration interact with spatiotemporal connectivity to affect metapopulation growth and stability. We used stochastic models and stage-structured metapopulation dynamics to study the interacting effects of ocean currents and dispersal traits on regional growth and stability. We derived stochastic metapopulation growth and stability, which predict the strong impact of local density regulation on the response of metapopulation to dynamic connectivity: temporal variance, positive (negative) covariance, and negative (positive) autocorrelation in connectivity deflate (inflate) density-independent growth. Yet, stability decreases (increases) with temporal variance and positive (negative) covariance of connectivity. We applied our derived metrics to simulated connectivity along the coast of British Colombia (Canada) over a range of spawning time (ST) and pelagic larval duration (PLD). Our analysis shows strong interactions between statistical components of connectivity, dispersal-related traits, and metapopulation growth and stability. The non-monotonic response of metapopulation stability to PLD was driven by mean of connectivity over short PLDs (< 36 days), with a decrease in mean connectivity and stability with PLD. Over longer PLDs, temporal variance in connectivity had a dominant effect, with both a decrease in temporal variance of connectivity and an increase stability with PLDs > 36 days. We therefore use a trait-based framework and partitioning the relative importance of spatial and temporal components of stochastic marine dispersal, to inform species response through climate-induced changes in larval transport and in dispersal-related traits.

Appendices

Appendix A: Analysis of density- independent metapopulation growth]

$$ \frac{dn_{i}}{dt}=r_{i} n_{i}+\sum\limits_{j \neq i}^{m} I_{ji}(t)n_{j}, $$

(10)

The system of equations can be formulated in matrix form:

$$ \frac{dN}{dt}=A(t)N, $$

(11)

where N(t) = (n₁(t),..., n_m(t))^T and A(t) = (a_{i, j})_{(i, j)} is the metapopulation projection matrix at time t, such that

$$\begin{array}{@{}rcl@{}} &&a_{ii}(t)=r_{i} i=i,\\ &&a_{ij}(t)=I_{ij}(t) i\neq j, \end{array} $$

(12)

The total abundance of the metapopulation is $N(t)={\sum }_{i = 1}^{m} n_{i}(t)$. In our model, the only fluctuating inputs are connectivity parameters, using a time incrementation large enough to assume that dynamics of continuous model in matrix form:

$$ \frac{dN}{dt}=A(t)N, $$

(13)

similar to discreet model

$$ N(t + 1)=(A(t)+I_{m})N(t), $$

(14)

where I_m is the identity matrix. If we let

$$ B(t)=A(t)+I_{m}, $$

(15)

and assuming that:

$$ \exists! \xi(t) \in \mathbb{R}^{m^{2}} / \left\{B(t)= \bar B +\xi(t), \mathbb{E}[\xi(t)]= 0; \forall t \in \mathbb{R^{+}}\right\}, $$

(16)

Assuming that average matrix $\bar B$ is a non-negative and irreducible and has m linearly independent left and right eigenvectors V _i and U_i and corresponding eigenvalues λ_i (i ∈{1,..., m} and λ_d = λ₁ > λ₂ > ... > λ_m), Tuljapurkar and Orzack (1980a, b) showed that the long-term stochastic growth follows a log normal distribution with a mean of λ_s and variance σ²:

$$ \log \lambda_{s} \simeq \log \lambda_{d} - \frac{\tau^{2}}{2{\lambda_{d}^{2}}}+\frac{{\Theta}_{1}}{{\lambda_{d}^{2}}}, $$

(17)

$$ \log \sigma^{2} =\frac{\tau^{2}}{{\lambda_{d}^{2}}} +\frac{2 {\Theta}_{2}}{{\lambda_{d}^{2}}}, $$

(18)

where

$$\begin{array}{@{}rcl@{}} \log(\lambda_{s})\!&=&\!\lim\limits_{t \rightarrow \infty} \frac{1}{t} \log \left\|\frac{N(t)}{N(0)}\right\| \\ \!&=&\! \lim\limits_{t \rightarrow \infty}\frac{1}{t} \log \left\|(A(t-1)+I_{m})\times ...\times (A(0)+I_{m}) \right\| \\ \!&=&\! \lim\limits_{t \rightarrow \infty}\frac{1}{t} \log \left\|B(t-1)\times ...\times B(0) \right\|, \end{array} $$

(19)

and

$$ \tau^{2}{}={}\frac{{\sum}_{k\neq l}{\sum}_{i\neq j} V_{1}(k)V_{1}(i)U_{1}(l)U_{1}(j) Cov(I_{kl}(t),I_{ij}(t))}{||V_{1} U_{1}||^{2}}, $$

(20)

$$ {\Theta}_{1}=\frac{{\sum}_{k\neq l}{\sum}_{i\neq j} {\sum}_{T = 1}^{\infty} {\sum}_{s = 1}^{m} V_{1}(k)U_{1}(l) \frac{\lambda_{s}}{\lambda_{d}}^{T-1}\frac{V_{s}(i)U_{s}(j) \text{Cov}(I_{kl}(t-T),I_{ij}(t))}{||V_{s}U_{s}||}}{||V_{1} U_{1}||} $$

(21)

$$ {\Theta}_{2}=\frac{{\sum}_{k\neq l}{\sum}_{i\neq j} {\sum}_{T = 1}^{\infty}V_{1}(k)V_{1}(i)U_{1}(l)U_{1}(j) \text{Cov}(I_{kl}(t-T),I_{ij}(t))}{||V_{1} U_{1}||^{2}} $$

(22)

$$ \tau^{2}=\sum\limits_{k\neq l}\sum\limits_{i\neq j} Z_{1kl}Z_{1ij}\text{Cov}(I_{kl}(t),I_{ij}(t)), $$

(23)

$$\begin{array}{@{}rcl@{}} {\Theta}_{1}&=&\sum\limits_{k\neq l}\sum\limits_{i\neq j}\sum\limits_{T = 1}^{\infty}\sum\limits_{s = 2}^{m} \left( \frac{\lambda_{s}}{\lambda_{d}}\right)^{T-1}Z_{1kl}Z_{sij}\text{Cov}\\&&\times\left( I_{kl}(t),I_{ij}(t-T)\right), \end{array} $$

(24)

$$ {\Theta}_{2}=\sum\limits_{k\neq l}\sum\limits_{i\neq j} \sum\limits_{T = 1}^{\infty}Z_{1kl}Z_{1ij}\text{Cov}\left( I_{kl}(t),I_{ij}(t-T)\right) $$

(25)

where $Z_{hij}=\frac {\partial \lambda _{h}}{\partial I_{ij}}=v_{hi} u_{hj}$, h ∈ 1,..., n, v_hi (u_hj) are the i th (j th) component of the left (right) eigenvector V _h (U_h) corresponding the h th eigenvalue (λ_d = λ₁ > λ₂ > ... > λ_n). For a metapopulation fragmented over m patches, there are m(m − 1) possible Var(I_ij(t)), and $\frac {(m(m-1)(m(m-1)-1))}{2}$ possible Cov(I_ij(t), I_kl(t)) terms. While Var(I_ij(t)) ≥ 0, the covariance Cov(I_ij(t), I_kl(t)) can be positive, negative, or 0.

Appendix B: Analysis of density-dependent stochastic growth

During the time interval [t, t + Δt], the probability of an increase by one individual larvae at patch i coming from patch j is:

$$\begin{array}{@{}rcl@{}} &&P((n_{1},...,n_{i}+ 1,...,n_{m})|(n_{1},...,n_{i},...,n_{m}))\\ &&=I_{ji}n_{j}{\Delta} t+O({\Delta}), \end{array} $$

(26)

However, the individual larvae can be dispersing from all other patches:

$$\begin{array}{@{}rcl@{}} &&P((n_{1},...,n_{i}+ 1,...,n_{m})|(n_{1},...,n_{i},...,n_{m}))\\ &&=\sum\limits_{j\neq i}^{m}I_{ji}n_{j}{\Delta} t+O({\Delta}), \end{array} $$

(27)

During a time interval [t, t + Δt], change in stochastic process is given by ΔX_t = (Δn_1t,...,Δn_mt). Each component of ΔX_t is the sum of its conditional expected value and a conditional centered random increment defined in (Table 2), where the conditioning is on X(t):

$$\begin{array}{@{}rcl@{}} {\Delta} n_{it}&=&E[{\Delta} n_{it}]+({\Delta} n_{it}-E[{\Delta} n_{it}])\\ &=&E[{\Delta} n_{it}]+\sum\limits_{j\neq i}^{m} {\Delta} T_{ji} -\sum\limits_{j\neq i}^{m} {\Delta} E_{ij} +{\Delta} R_{i}-{\Delta} D_{i}, \end{array} $$

For example, the increment ΔT_ji represents the j th conditionally centered locally Poisson increment $I_{ji} n_{j} \left (1-\frac {n_{i}}{K_{i}} \right ){\Delta } t$ that is related to the change in the i th components of X(t).

For sufficiently large initial total density N(t₀), we can replace centered Poisson increments ΔT_ji by $G^{t}_{ji} {\Delta } W_{ji}$, for i, j = 1, 2,..., m, where each $G^{t}_{ji}\sqrt {{\Delta } t}$ is the approximate conditional standard deviation of the increment of respective transition and each W_ij is a standard Wiener process, that is, ${\Delta } W_{ij} \sim \mathcal {N}(0,{\Delta } t)$ (Kurtz 1978), and $G_{ij}^{t}= \sqrt {I_{ji}\frac {n_{j}}{N}(1-\frac {n_{i}}{K_{i}})}$ or $G_{ji}^{t}= \sqrt {I_{ji}\frac {n_{j}}{N}(1-\frac {n_{i}}{K_{i}})}$. The Markov process x(t) = (x₁(t),..., x_m(t)) is approximated by the m-dimensional diffusion satisfying the system of stochastic differential equations (Kurtz 1978, 1981) with the initial distribution given by:

$$ dx_{it}=\left( \left( r_{i}x_{it}+\sum\limits_{j\neq i}I_{ji} x_{jt}\right)\right.\left( 1-\frac{x_{i}}{k_{i}}\right)dt + \sum\limits_{j\neq i} G^{t}_{ji} dW^{t}_{ji} $$

(28)

Ethier and Kurtz (2009) showed that the normalized diffusion process $\sqrt {N }(x_{1t}-k_{1},...,x_{mt}-k_{m})$ is approximated near the stable equilibrium (0,..., 0) by the diffusion process:

$$ dV_{t}=J_{0} V_{t}+G^{t} dW^{t}, $$

(29)

where V _t = (x_1t − k₁,..., x_mt − k_m)^T, with local drift J₀ the Jacobian matrix of the scaled deterministic system (28) at equilibrium (k₁,..., k_m)), and the diffusion matrix $G^{t}=(G_{ij}^{t})_{(i,j)}$.

$$ \left( \begin{array}{c} dv_{1}\\ .\\ .\\ .\\ dv_{m} \end{array}\right) =J_{0}\left( \begin{array}{c} v_{1}\\ .\\ .\\ .\\ v_{m} \end{array}\right) + G^{t}\left( \begin{array}{c} dW_{21}\\ .\\ .\\ .\\ dW_{m1}\\ dW_{12}\\ .\\ .\\ .\\ dW_{m2}\\ .\\ .\\ .\end{array}\right) $$

(30)

where

$$ J_{0}=\left( \begin{array}{cccccc} - r_{1}-{\sum}_{j\neq 1}\bar I_{j1} \frac{k_{j}}{k_{1}} & 0&.&.&.&0 \\ 0&.&.&.&.&0\\ .&.&.&.&.&.\\ .&.&.&.&.&.\\ .&.&.&.&.&.\\ 0&.&.&.&0& -r_{m}-{\sum}_{j\neq m}\bar I_{jm} \frac{k_{j}}{k_{m}} \end{array}\right) $$

(31)

and

$$\begin{array}{@{}rcl@{}} G^t &=& \left( \begin{array}{ccccccccccccccc} G^t_{21} &G^t_{31}& .&.&G^t_{m1}&-G^t_{12}&-G^t_{13}&.&. &-G^t_{1m}&{\Delta} R_1&-{\Delta} D_1&0\\ -G^t_{21}&0&.&.&0 & G^t_{12} &0 &.&.&.&.&.&.\\ 0&-G^t_{31}&0&.&.& 0&G^t_{13}&0&.&.&.&.&.\\ .&.&-G^t_{41}&.&.& .&. &G^t_{41}&.&.&.&.&.\\ .&.&.&.&.& .&. &.&.&.&.&.&.\\ .&.&.&.&.&-G^t_{m1}&. &.&.&G^t_{1m}&.&.&. \end{array}\right.\\ &&\left.\begin{array}{ccccccccccccccccccc} .&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&0\\ G^t_{32}&G^t_{42}&.&.&.&G^t_{m2}&-G^t_{21}&-G^t_{23}&.&.&.&-G^t_{2m}&{\Delta} R_2&-~{\Delta} D_2&0&.&.&.&.\\ -G^t_{32}&0&.&.&.&.&.&G^t_{23}&.&.&. &.&.&.&.&.&.&.&.\\ 0&-G^t_{42}&0&.&.&.&.&.& G^t_{24}&. &.&.&.&.&.&.&. &.&.\\ .&.&.&.&.&.&.&.& .&. &.&.&.&.&.&.&. &0&0\\ .&.&.&.&.&.&.&-G^t_{m2}& .&. &.&G^t_{2m}&.&.&..&. &{\Delta} R_m&-{\Delta} D_{m} \end{array}\right) \end{array} $$

(32)

The stationary distribution of this OU process is Gaussian with mean zero ((0,.., 0)) and covariance $S=(\sigma ^{2}_{ij})_{(i,j)\in \{1,...,m\}^{2}}$ given by the relation (Andersson and Britton 2000)

$$J_{0} S +S {J_{0}^{T}} =-C, $$

The matrix C = (c_ij)_{(i, j)} elements are obtained from matrix G^t.

$$\begin{array}{@{}rcl@{}} c_{ii}(t){}&=&{}\sum\limits_{q\neq i}^{m}\sum\limits_{p\neq i}^{m} k_{i}k_{q}Cov(I_{qi}(t),I_{ip}(t))+\epsilon {\kern15pt} i=i \end{array} $$

(33)

$$\begin{array}{@{}rcl@{}} c_{ij}(t)&=&k_{i}k_{j}Cov(I_{ij}(t),I_{ji}(t))+\epsilon \qquad \qquad {\kern6pt} i\neq j \end{array} $$

(34)

where the 𝜖 is of order $O(min\{{k^{2}_{i}}, k\in (1,...,n)\})$.

Appendix C: Connectivity statistical mean, variance, covariance, and auto-correlations

Fig. 6

The mean of mean connectivity distribution $\{\mu (C_{ij}(t))\}_{(i,j)\in \{1,...,388\}^{2}}$ as a function of larval duration and spawning time

Fig. 7

The mean of variance of connectivity distribution $\{\sigma ^{2}(C_{ij}(t))\}_{(i,j)\in \{1,...,388\}^{2}}$ as a function of larval duration and spawning time

Fig. 8

The mean of connectivity covariance distribution $\{cov(C_{ij}(t),C_{kl}(t))\}_{(i,j,k,l)\in \{1,...,388\}^{4}}$ as a function of larval duration and spawning time

Fig. 9

Temporal connectivity autocorrelation and how they vary with larval duration. The z -axis represent the mean of $\{ cov(C_{ij}(t),C_{kl}(t-T))\}_{(i,j,k,l,T) \in \{1,...,388\}^{4} X \{1,...,9\}}$