Homing fidelity and reproductive rate for migratory populations

Abstract

Short-term and long-term population growth rates can differ considerably. While changes in growth rates can be driven by external factors, we consider another source for changes in growth rate. That is, changes are generated internally by gradual modification of population structure. Such a modification of population structure may take many generations, particularly when the populations are distributed spatially in heterogeneous environments. Here, the net reproductive rate R ₀ is not sufficient to characterize short-term growth. Indeed, a population with net reproductive rate greater than one could initially decline precipitously, or a population with net reproductive rate less than one could initially grow substantially. Thus, we augment the net reproductive rate with lower and upper bounds for the transient reproductive rate, R _l and R _u . We apply these measures to the study of spatially structured salmon populations and show the effect of variable homing fidelity on short-term and long-term generational growth rates.

Appendices

Appendix A: Graph reduction method

This method starts with the description of the projection matrix as a life cycle graph. Once the life cycle graph has been specified, the calculation procedure is as follows. (1) Identify survivorship and fecundity transitions. (2) Multiply all fecundity transitions in the graph by $R^{-1}_{0}$.(3) Eliminate survivorship self-loops, using rule a in Fig. 10. (4) Reduce the graph using the graph reduction rules defined in Figure I until only nodes with fecundity self-loops are left. When a node is eliminated, all pathways that go through that node have to be recalculated. (5) If only one node with a single self-loop is left, eliminate the final node by setting the self-loop equal to 1 and solve this equation for R ₀.

Fig. 10

Graph reduction rules. a self-loop elimination with b < 1. b Parallel path elimation. c Node elimination. Rules a and b show elimination of paths, and rule c shows the elimination of node 2. Graph reduction is done by repeatedly applying these rules until only nodes are left

Appendix B: Proof of (1.6)

To investigate the function P x, we let P = (p _{i j} ) _n×n and x = [x ₁, x ₂, ⋯ , x _n ]^T. A simple calculation gives

$$ ||\mathbf{P}x||_{1}=\sum\limits_{i=1}^{n}p_{i1}x_{1}+\sum\limits_{i=1}^{n}p_{i2}x_{2}+\cdots+ \sum\limits_{i=1}^{n}p_{in}x_{n}, $$

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Noticing that $\sum \limits _{i=1}^{n}p_{ij} (j=1, \cdots , n)$ is the j ^th column sum of the matrix P, we consider the smallest such column sum of P. Suppose that for some 1 ≤ k ≤ n, $\sum \limits _{i=1}^{n}p_{ik}=\min _{1\leq j\leq n}\sum \limits _{i=1}^{n}p_{ij}$, then the function ||P x||₁ has minimum value $\sum \limits _{i=1}^{n}p_{ik}$ when x is a unit vector with x _k = 1 and x _j = 0 (j ≠ k). That is to say,

$$ \lambda_{l}=\min\limits_{1\leq j\leq n}\sum\limits_{i=1}^{n}p_{ij}, $$

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the minimum sum of column vectors of projection matrix P.

Appendix C: R ₀ for resident salmonids

In terms of the life cycle of resident species, we divide the population into four groups: fertilized egg (E), fry (F), juvenile (J), and adult (A). We take time unit to be 1 year. The population vector is x(t) = [E(t), F(t), J(t), A(t)]^T, which represents the population density of each stage at the end of the breeding season in year t. We relate the population density of each stage at time t + 1 to time t by the matrix equation

$$ x(t+1) = \mathbf{P}x(t), $$

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where the projection matrix P is

$$ \mathbf{P}=\left( \begin{array}{cccc} 0 & 0 & 0& b \\ p_{fe} & 0 & 0 & 0 \\ 0 & p_{jf} & p_{jj}& 0\\ 0 & 0& p_{aj}& p_{aa} \end{array} \right). $$

Here, b is the average number of fertilized eggs produced per adult per year, p _{f e} is the proportion of eggs that hatch to fry stage each year, p _{j f} is the proportion of fry that survive to the juvenile stage each year, p _{j j} is the proportion of juveniles that survive to remain as a juvenile per year, p _{a j} is the proportion of juveniles that survive to become adults each year, p _{a a} is the proportion of adults that survive each year. The vital rates of salmonids living a variety of environment have been estimated by many researchers (e.g., (Al-Chokhachy and Budy 2008; Bowerman and Budy 2012; McPhail and Baxter 1996)).

The matrix Eq. 26 models the dynamics of resident salmonids population. The net reproductive rate, R ₀, for this population can be calculated using the graph reduction method, as mentioned in Appendix A. The graph reduction method is shown in Fig. 11.

Fig. 11

a The full transformed graph. b Eliminating self-loops. c Eliminating nodes F, J, and A. d Solving for R ₀

From the equation of R ₀ (Fig. 11d), we see that the proportion of individuals that start as eggs and eventually mature and survive to become breeding adults is p _{f e} p _{j f} p _{a j} /[(1−p _{j j} )(1−p _{a a} )], and the expected number of eggs produced per breeding adult is b. Multiplying these quantities yields R ₀.

Appendix D

Fig. 12

b Eliminating self-loops. c Eliminating node J ₁. d Eliminating nodes F ₁, J ₂, and A ₂

Appendix E

Fig. 13

b Eliminating self-loops. c Eliminating nodes E ₁, E ₃, A ₂ and A ₄. d Eliminating nodes J ₁ and J ₃. e Eliminating nodes J ₂ and J ₄. f Eliminating nodes F ₃

Appendix F: An analytical calculation of ∂ R/∂ 𝜖 when 𝜖 is close to 0 and 1

We differentiate the Eq. 15 with respect to 𝜖 to get

$$\begin{array}{@{}rcl@{}} &&2R_{0}\frac{\partial R_{0}}{\partial\epsilon}+[2A(1-\epsilon)-2B\epsilon]R_{0}- \left[A(1-\epsilon)^{2}+B\epsilon^{2}\right]\frac{\partial R_{0}}{\partial\epsilon}\\ &&-C\left(2\epsilon-6\epsilon^{2}+4\epsilon^{3}\right) +4D\epsilon^{3}-4E(1-\epsilon)^{3}=0. \end{array} $$

Thus,

$$ \frac{\partial R_{0}}{\partial\epsilon} =\frac{2AR_{0}(\epsilon-1)+2BR_{0}\epsilon+C\left(2\epsilon-6\epsilon^{2}+4\epsilon^{3}\right) -4D\epsilon^{3}+4E(1-\epsilon)^{3}}{2R_{0}-A(1-\epsilon)^{2}-B\epsilon^{2}}. $$

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Hence,

$$ \lim\limits_{\epsilon\rightarrow 0^{+}}\frac{\partial R_{0}}{\partial\epsilon}=\frac{-2AR_{0}\left(0^{+}\right)+4E}{2R_{0}(0^{+})-A}. $$

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Noticing that R ₀(0⁺) = max{G ₂₁ G ₁₂, G ₄₃ G ₃₄}, we find

$$\begin{array}{@{}rcl@{}} \lim\limits_{\epsilon\rightarrow 0^{+}}\frac{\partial R_{0}}{\partial\epsilon}&=&\frac{-2(G_{21}G_{12}+G_{43}G_{34}) \max\{G_{21}G_{12}, G_{43}G_{34}\} +4G_{21}G_{12}G_{43}G_{34}}{2\max\{G_{21}G_{12}, G_{43}G_{34}\}-G_{21}G_{12}-G_{43}G_{34}}\\ &=&-2\max\{G_{21}G_{12}, G_{43}G_{34}\}. \end{array} $$

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Since R ₀(1⁻) = max{G ₂₃ G ₃₂, G ₄₁ G ₁₄}, similar computation yields

$$\begin{array}{@{}rcl@{}} \lim\limits_{\epsilon\rightarrow 1^{-}}\frac{\partial R_{0}}{\partial\epsilon}=2 \max\{G_{23}G_{32}, G_{41}G_{14}\}. \end{array} $$

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If 0 < 𝜖 ≪ 1, the function R ₀(𝜖) can be approximated by a straight line

$$\begin{array}{@{}rcl@{}} R_{0}(\epsilon)&\approx& R_{0}\left(0^{+}\right)+\left(\lim\limits_{\epsilon\rightarrow 0^{+}}\frac{\partial R_{0}}{\partial\epsilon}\right)\epsilon\\ &=&\max\{G_{21}G_{12}, G_{43}G_{34}\}\\ &&-2\max\{G_{21}G_{12}, G_{43}G_{34}\}\epsilon, \end{array} $$

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with negative slope.

Similarly, if 𝜖 is less than and sufficiently close to 1, then we have

$$\begin{array}{@{}rcl@{}} R_{0}(\epsilon)&\approx& R_{0}(1^{-})+\left(\lim\limits_{\epsilon\rightarrow 1^{-}}\frac{\partial R_{0}}{\partial\epsilon}\right)(\epsilon-1)\\ &=&-\max\{G_{23}G_{32}, G_{41}G_{14}\}\\ &&+2\max\{G_{23}G_{32}, G_{41}G_{14}\}\epsilon. \end{array} $$

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with positive slope.

Appendix G: Further discussion about R _l and R _u

For the matrix Q, we define the range of Q as Ran$(\mathbf {Q}) = \{\mathbf {Q}x|x\in \mathbb {R}^{n}\}$, and the null space of Q as $\mathcal {N}(\mathbf {Q}) = \{x\in \mathbb {R}^{n}|\mathbf {Q}x=0\}$.

Then both Ran (Q) and $\mathcal {N}(\mathbf {Q})$ are subspaces of $\mathbb {R}^{n}$, and $\mathbb {R}^{n}=\text {Ran}(\mathbf {Q})\bigoplus \mathcal {N}(\mathbf {Q})$. Therefore, for any $x\in \mathbb {R}^{n},$ there exists unique x _Ran ∈ Ran(Q) and unique $x_{\mathcal {N}}$ such that $x=x_{\text {Ran}}+x_{\mathcal {N}}$. Thus, for any $x\in \mathbb {R}^{n},$ $\mathbf {Q}x=\mathbf {Q}x_{\text {Ran}}+\mathbf {Q}x_{\mathcal {N}}=\mathbf {Q}x_{\text {Ran}}+0$. The projection of any $x\in x_{\mathcal {N}}$ will give zero individual in the next generation, which is not of biological interest. For this reason, we restrict x ∈ Ran(Q) when defining R _l and R _u .

Appendix H: A multiple-patch model

If we assume that there are I small rivers in upstream and K big rivers in downstream, it is not difficult to extend the four-patch model (12) to a multiple-patch model:

$$\begin{array}{@{}rcl@{}} E_{i}(t+1)&=&\sum\limits_{k=1}^{K} b_{i}m_{a}^{ik}A_{k}(t)\\ F_{i}(t+1)&=&p_{fe}^{i}E_{i}(t)\\ J_{i}(t+1)&=&\left(1-\sum\limits_{k=1}^{K}m_{f}^{ki}\right)p_{jf}^{i}F_{i}(t) +\left(1-\sum\limits_{k=1}^{K}m_{j}^{ki}\right)p_{jj}^{i}J_{i}(t)\\ J_{k}(t+1)&=&\sum\limits_{i=1}^{I}\left(m_{f}^{ki}p_{jf}^{ki}F_{i}(t)+ m_{j}^{ki}p_{jj}^{ki}J_{i}(t)\right)\\ A_{k}(t+1)&=&p_{aj}^{k}J_{k}(t)+p_{aa}^{k}A_{k}(t), \end{array} $$

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for i = 1, 2, ⋯ , I and k = 1, 2, ⋯ , K.