On fitness in metapopulations that are both size- and stage-structured

Abstract

A proxy for the invasion fitness in structured metapopulation models has been defined as a metapopulation reproduction ratio, which is the expected number of surviving dispersers produced by a mutant immigrant and a colony of its descendants. When a size-structured metapopulation model involves also individual stages (such as juveniles and adults), there exists a generalized definition for the invasion fitness proxy. The idea is to calculate the expected numbers of dispersers of all different possible types produced by a mutant clan initiated with a single mutant, and to collect these values into a matrix. The metapopulation reproduction ratio is then the dominant eigenvalue of this matrix. The calculation method has been published in detail in the case of small local populations. However, in case of large patches the previously published numerical calculation method to obtain the expected number of dispersers does not generalize as such, which gives us one aim of this article. Here, we thus derive a generalized method to calculate the invasion fitness in a metapopulation, which consists of large local populations, and is both size- and stage-structured. We also prove that the metapopulation reproduction ratio is well-defined, i.e., it is equal to 1 for a mutant with a strategy equal to the strategy of a resident. Such a proof has not been previously published even for the case with only one type of individuals.

Appendix: Cyclic case

In the main text we considered only the case, in which the local population sizes approach an equilibrium, when $\tau \rightarrow \infty$. In a stage-structured model, however, this is not necessarily the case. Instead, population sizes may approach, e.g., a limit cycle. In that case we calculate Eqs.  (12) and  (26) until time T so that the population sizes are close enough to a point on the limit cycle. After this we use the approximation $\mathbf {X}(T + k\varDelta T + \tau )=\mathbf {X}(T + \tau )$ for all $0 \leqslant \tau \leqslant \varDelta T$ and $k=0,1,2,\ldots$, where $\varDelta T$ is the length of the limit cycle.

First we denote $t=T + k\varDelta T + \tau$. Then we can write the expression for $\mathcal {F}$ corresponding to (14)

$$\begin{aligned} \mathcal {F}(t)= & {} \mathcal {F}(T)\exp \left( -\int _{T}^{t}\delta (\mathbf {X}(s))ds\right) \nonumber \\= & {} \mathcal {F}(T)\exp \left( -\int _{T}^{T+\varDelta T}\delta (\mathbf {X}(s))ds\right) ^{k}\exp \left( -\int _{T}^{T+\tau }\delta (\mathbf {X}(s))ds\right) . \end{aligned}$$

(31)

By using an auxiliary variable $z=\exp (-\int _{T}^{T+\varDelta T}\delta (\mathbf {X}(s))ds)$ we can write

$$\begin{aligned} \mathbf {E}_i(\infty )= & {} \mathbf {E}_i(T)+\int _{T}^{\infty }\varvec{\rho } (\mathbf {X}(\tau ),s_i)\mathbf {X}_i(t)\mathcal {F}(t)dt \nonumber \\= & {} \mathbf {E}_i(T)\nonumber \\&+\,\mathcal {F}(T)\sum _{k=0}^{\infty }z^k\int _{T}^{T+\varDelta T}\varvec{\rho }(\mathbf {X}(\tau ),s_i) \mathbf {X}_i(\tau ) \exp \left( -\int _{T}^{T+\tau }\delta (\mathbf {X}(s))ds\right) d\tau \nonumber \\= & {} \mathbf {E}_i(T) +\frac{\mathcal {F}(T)}{1-z} \int _{T}^{T+\varDelta T}\varvec{\rho }(\mathbf {X}_i(\tau ),s_i) \mathbf {X}_i(\tau ) \nonumber \\&\exp \left( -\int _{T}^{T+\tau }\delta (\mathbf {X}(s))ds\right) d\tau \end{aligned}$$

(32)

and

$$\begin{aligned} L(\infty )= & {} L(T)+\int _{T}^{\infty } \mathcal {F}(t)dt \nonumber \\= & {} L(T) +\frac{\mathcal {F}(T)}{1-z} \int _{T}^{T+\varDelta T} \exp \left( -\int _{T}^{T+\tau }\delta (\mathbf {X}(s))ds\right) d\tau \end{aligned}$$

(33)

which correspond to Eq.  (15).

Similarly as above, we obtain the invasion fitness (proxy) in a cyclic case by rewriting the emigrant production $E_j^\mathrm{prod}$ in the Eq. (27)

$$\begin{aligned} E_{j}^{\mathrm{prod}}(\hat{\mathbf {Y}})= & {} \frac{\pi }{L}\int _{0}^{\infty }\mathcal {F}(t)\rho _{j}(\mathbf {X}(t),s_\mathrm{mut})N_j(t)dt\nonumber \\= & {} \frac{\pi }{L(\infty )}\bigg [Q_{j}(T) +\frac{\mathcal {F}(T)}{1-z} \int _{T}^{T+\varDelta T} \rho _{j}(\mathbf {X}(\tau ),s_\mathrm{mut}) N_{j}(\tau ) \nonumber \\&\exp \bigg (-\int _{T}^{T+\tau }\delta (\mathbf {X}(s))ds\bigg ) d\tau \bigg ]. \end{aligned}$$

(34)