Diffusion model for initial colonization of Spartina patches on Korean tidal flats

Abstract

Spartina alterniflora Loisel, widely recognized as an aggressive invader of estuaries and salt marshes worldwide, was recently reported in Korean waters as rapidly invading intertidal mudflats, growing in circular patches. For more effective control management of the invasive cordgrass, we developed a modified ignition logistic model based on the satellite imageries to estimate the settlement time of the first individual stand and the doubling period of the patch spread in the early colonization state. The present model is designed for estimating the starting time and the doubling period of the patch spread at the growth stage, which is a salient feature different from other logistic models. The importance of estimating the starting time of the invasion may lie in figuring out what ecological changes occurred at that time. A Monte-Carlo simulation is combined with our model to obtain reliable predictions against the noisy data. As a result of applying the model to the Northwest Pacific invading tidal flats of Korea, China, and Japan, it turns out that the doubling periods of the patch spread are generally shorter and similar to each other, which range from 0.6 to 2.1 years. This is probably attributed to the genetically hybridized populations of Spartina alterniflora invading this region.

Appendices

Appendix A: Derivation of the ignition logistic function

For the solution of the ignition logistic model defined in (7) with initial condition (8), we start from the separation of variables of the differential equation,

$$\begin{aligned} \int _0^A \frac{1}{\alpha \,\eta (A^* - \eta )+\beta }\,d\eta = \int _{t_0}^t 1\,dt, \end{aligned}$$

(A1)

because of the initial condition that $A = 0$ when $t = t_0$. Using the definition of $A^\pm$ in (11), the left hand side of (A1) can be calculated to reach

$$\begin{aligned} \int _0^A \frac{1}{\alpha \,\eta (A^* - \eta )+\beta }\,d\eta&= -\,\frac{1}{\alpha \,\sqrt{\Delta }} \int _0^A \frac{1}{\eta -A^+} - \frac{1}{\eta +A^-}\,d\eta \nonumber \\ {}&= \frac{1}{\alpha \,\sqrt{\Delta }} \log \frac{1+\frac{A}{A^-}}{1-\frac{A}{A^+}}. \end{aligned}$$

(A2)

Considering the left hand side of (A1), we reach the following identity,

$$\begin{aligned} \log \frac{1+\frac{A}{A^-}}{1-\frac{A}{A^+}} = \alpha \,\sqrt{\Delta }\,(\, t - t_0 \,). \end{aligned}$$

(A3)

Solve this equation for the variable A and we finally obtain the ignition logistic function written in (9).

Appendix B: The correspondence in Table 1

Comparing (9) and (10) with (12) and (13), respectively, we find the following four relationships, where $\delta _0$, $\delta _1$, $\delta _2$, and $\delta _3$ are parameters determined from the fitting model:

$$\begin{aligned} A^+&= \delta _3, \end{aligned}$$

(B4)

$$\begin{aligned} \frac{A^-}{A^- + A^+}&= \delta _2, \end{aligned}$$

(B5)

$$\begin{aligned} \alpha \,( A^- + A^+ )&= \frac{\delta _0}{t_R}, \end{aligned}$$

(B6)

$$\begin{aligned} - \,\alpha \,( A^- + A^+ )\,t_0&= -\,\frac{t_c}{t_R}\,\delta _0 + \log \delta _1, \end{aligned}$$

(B7)

where the values, $t_c$ and $t_R$, are determined from the data in terms of (15). From equations (B4) and (B5), we infer that $A^- = \delta _2\,\delta _3/(1-\delta _2)$ and $A^- + A^+ = \delta _3/(1-\delta _2)$. Therefore, substituting these values into equation (B6), we obtain

$$\begin{aligned} \alpha = \frac{\delta _0\,(1-\delta _2)}{\delta _3\,t_R}. \end{aligned}$$

Eliminating the quantity $\alpha (A^- + A^+)$ from equations (B6) and (B7), we find

$$\begin{aligned} t_0 = t_c - \frac{t_R}{\delta _0}\,\log \delta _1. \end{aligned}$$

Given the identity $\Delta = (A^*)^2 + 4 \beta /\alpha$, and knowing that $A^- + A^+ = \sqrt{\Delta }$ and $A^+ - A^- = A^*$ from (11), we deduce that $\beta = \alpha \,A^- A^+$. As a result, we have

$$\begin{aligned} \beta&= \frac{\delta _0\,\delta _2\,\delta _3}{t_R}, \\ A^*&= \frac{\delta _3\,(1-2\delta _2)}{1-\delta _2}. \end{aligned}$$

Meanwhile, let us estimate the initial doubling period $T > 0$. For a sufficiently small time just after $t_0$, E(t) in (10) is small enough to have the following asymptotic behavior of the ignition logistic function $A^{ILM}(t)$ in (9):

$$\begin{aligned} A^{ILM}(t)&= A^+\,E(t) + \mathcal {O}(E^2) \nonumber \\ {}&= \frac{\beta }{\alpha \,(A^-+A^+)} \left( \, e^{\alpha \,(A^-+A^+)\,(\,t - t_0\,)} -1 \,\right) + \mathcal {O}(E^2), \end{aligned}$$

(B8)

as E(t) goes to zero. Comparing the leading term of (B8) with that of (4), the following relationship should be satisfied:

$$\begin{aligned} \frac{\log 2}{T} = \alpha \,(A^- + A^+). \end{aligned}$$

(B9)

Based on the identity that $\alpha \,(A^-+A^+) = {\delta _0}/{t_R}$, it is reasonable to define the initial doubling period T in the ILM as follows:

$$\begin{aligned} T \equiv \frac{t_R}{\delta _0}\,\log 2. \end{aligned}$$

(B10)

Appendix C: Error bound in (21)

Using $\vert \partial D_i \vert = 2 \sqrt{\pi } \sqrt{\vert D_i \vert }$, the summation on the right-hand side of (20) is addressed as

$$\begin{aligned} \sum _{i=1}^{N^{p}} \vert \partial D_i \vert \times \left( \sqrt{2} \sqrt{\blacksquare } \right) = 2 \sqrt{2 \pi } \sqrt{\blacksquare } \, \sum _{i=1}^{N^{p}} \sqrt{ \vert D_i \vert } . \end{aligned}$$

(c11)

Indeed, based on the inequality $\sqrt{\vert D_i \vert }\, \sqrt{\vert D_j \vert } \le \frac{1}{2} \left( \vert D_i \vert + \vert D_j \vert \right)$, the direct calculation yields

$$\begin{aligned} \left( \sum _{i=1}^{N^p} \sqrt{\vert D_i \vert } \right) ^2&= \sum _{i=1}^{N^p} \sum _{j=1}^{N^p} \sqrt{\vert D_i \vert }\, \sqrt{\vert D_j \vert } \\ {}&\le \sum _{i=1}^{N^p} \sum _{j=1}^{N^p}\frac{1}{2} \left( \vert D_i \vert + \vert D_j \vert \right) \\ {}&= \frac{1}{2} \left( \sum _{j=1}^{N^p} \sum _{i=1}^{N^p} \vert D_i \vert + \sum _{i=1}^{N^p} \sum _{j=1}^{N^p} \vert D_j \vert \right) = \vert D \vert \times N^p. \end{aligned}$$

Here, the last equality comes from the fact that $\sum _{i=1}^{N^p} \vert D_i \vert = \sum _{j=1}^{N^p} \vert D_j \vert = \vert D \vert$, which is based on the property of disjoint union. It is worth noting that the inequality in (21) holds even if $D = \bigcup _{i=1}^{N^p}$ is not a disjoint union.