Spatiotemporal behavior in a predator–prey model with herd behavior and cross-diffusion and fear effect

Abstract

In this manuscript, we investigate the fear effect on the spatiotemporal behavior of the predator–prey model with prey social behavior and cross-diffusive. Our main interest is to determine the existence of Turing patterns and the fear effect performed by predators on the prey population and the group defense. The fear can lead to a partition of the prey herd which is known as prey escaping. It is obtained that the system has rich dynamics elaborated by the presence of Turing patterns and Turing–Hopf bifurcation. The nature of Turing patterns is successfully discussed by analyzing the amplitude equations with a multiple-timescale technique. After studying the stability of these amplitude equations, it has been identified various Turing patterns driven by the cross-diffusion. Further, the effect of the fear rate or escaping rate on the behavior of the solution is discussed. The theoretical results are checked numerically.

Appendix A. The expressions of the terms of the amplitude equations (3.11)

First, we give the expressions of $f^{(1)}_{N_{1}N_{1}},\ldots ,f^{(1)}_{N_{2}N_{2}N_{2}}$ and $f^{(2)}_{N_{1}N_{1}},\ldots ,f^{(2)}_{N_{2}N_{2}N_{2}}$ of (3.1) as

$$\begin{aligned}&f^{(1)}_{N_{1}N_{1}}=-2\dfrac{r}{k}+\alpha (1-\alpha ) \beta ^{\alpha } (1-\eta )^{\alpha }(N_{1}^{*})^{\alpha -2}N_{2}^{*},\\&\quad f^{(1)}_{N_{1}N_{2}}=-\alpha \beta ^{\alpha }(1-\eta )^{\alpha }(N_{1}^{*})^{\alpha -1}-\gamma \eta ,\quad f^{(1)}_{N_{2}N_{2}}=0, \\&f^{(1)}_{N_{1}N_{1}N_{1}}=\alpha (1-\alpha )(\alpha -2) \beta ^{\alpha } (1-\eta )^{\alpha }(N_{1}^{*})^{\alpha -3}N_{2}^{*},\\&\quad f^{(1)}_{N_{1}N_{1}N_{2}}=\alpha (1-\alpha ) \beta ^{\alpha } (1-\eta )^{\alpha }(N_{1}^{*})^{\alpha -2},\quad f^{(1)}_{N_{2}N_{1}N_{2}}=0,\\&f^{(1)}_{N_{2}N_{2}N_{2}}=0,\quad f^{(2)}_{N_{2}N_{2}}=0,\\&\quad f^{(2)}_{N_{1}N_{2}}=e\alpha \beta ^{\alpha } (1-\eta )^{\alpha }(N_{1}^{*})^{\alpha -1}+e\gamma \eta ,\quad f^{(2)}_{N_{1}N_{1}}=e\alpha (\alpha -1) \beta ^{\alpha } (1-\eta )^{\alpha }(N_{1}^{*})^{\alpha -2}N_{2}^{*} \\&f^{(2)}_{N_{1}N_{1}N_{1}}=e\alpha (\alpha -1)(\alpha -2) \beta ^{\alpha } (1-\eta )^{\alpha }(N_{1}^{*})^{\alpha -3}N_{2}^{*},\\&\quad f^{(2)}_{N_{1}N_{2}N_{2}}=0,\quad f^{(2)}_{N_{2}N_{2}N_{2}}=0. \end{aligned}$$

Next, by using the standard method of multiple timescales developed in [26, 27], we try to extract the coefficients of amplitude equations (3.11) of the cross-diffusion model (1.2) near the onset $\delta _{21}=\delta _{21}^\mathrm{T}$. Taking the multiple timescales $t_{0}=t, \ t_{1}=\epsilon t, \ t_{2}=^{2}\epsilon t$, then we obtain

$$\begin{aligned} \dfrac{\partial }{\partial t}=\dfrac{\partial }{\partial t_{0}}+\epsilon \dfrac{\partial }{\partial t_{1}}+\epsilon ^{2}\dfrac{\partial }{\partial t_{2}}+O(\epsilon ^{2}). \end{aligned}$$

(A1)

From Eq. (3.1), if we destabilize the coefficients of $\epsilon ^{i}$, we get

$$\begin{aligned} L_{T}\left( \begin{array}{c} N_{1}^{(1)}\\ N_{2}^{(1)} \end{array} \right) =\left( \begin{array}{c} 0\\ 0 \end{array} \right) . \end{aligned}$$

(A2)

At the perturbation $O(\epsilon ^{2})$, we have

$$\begin{aligned} L_{T}\left( \begin{array}{c} N_{1}^{(2)}\\ N_{2}^{(2)} \end{array} \right) =&\dfrac{\partial }{\partial t_{1}}\left( \begin{array}{c} N_{1}^{(1)}\\ N_{2}^{(1)} \end{array} \right) -\delta _{21}^{(1)}J\left( \begin{array}{c} N_{1}^{(1)}\\ N_{2}^{(1)} \end{array} \right) \nonumber \\&-\dfrac{1}{2}\left( \begin{array}{lc} f^{(1)}_{N_{1}N_{1}}\left( N_{1}^{(1)}\right) ^{2}+2f^{(1)}_{N_{1}N_{2}}N_{1}^{(1)}N_{2}^{(1)}+f^{(1)}_{N_{2}N_{2}}\left( N_{2}^{(1)}\right) ^{2}\\ f^{(2)}_{N_{1}N_{1}}\left( N_{1}^{(1)}\right) ^{2}+2f^{(2)}_{N_{1}N_{2}}N_{1}^{(1)}N_{2}^{(1)}+f^{(2)}_{N_{2}N_{2}}\left( N_{2}^{(1)}\right) ^{2} \end{array} \right) \triangleq \left( \begin{array}{c} F_{N_{1}}\\ F_{N_{2}} \end{array} \right) , \end{aligned}$$

(A3)

with a similar process, we can obtain

$$\begin{aligned} L_{T}\left( \begin{array}{c} N_{1}^{(3)}\\ N_{2}^{(3)} \end{array} \right)= & {} \left( \begin{array}{c} \dfrac{\partial N_{1}^{(1)}}{\partial t_{2}}+\dfrac{\partial N_{1}^{(2)}}{\partial t_{1}}\\ \dfrac{\partial N_{2}^{(1)}}{\partial t_{2}}+\dfrac{\partial N_{2}^{(2)}}{\partial t_{1}} \end{array} \right) -\delta _{21}^{(1})J\left( \begin{array}{c} N_{1}^{(2)}\\ N_{2}^{(2)} \end{array} \right) -\delta _{21}^{1}J\left( \begin{array}{c} N_{1}^{(1)}\\ N_{2}^{(1)} \end{array} \right) \nonumber \\&-\left( \begin{array}{lc} f^{(1)}_{N_{1}N_{1}}N_{1}^{(1)}N_{1}^{(2)}+f^{(1)}_{N_{1}N_{2}}\left( N_{1}^{(1)}N_{2}^{(2)}+N_{1}^{(2)}N_{2}^{(1)}\right) +f^{(1)}_{N_{2}N_{2}}N_{2}^{(1)}N_{2}^{(2)}\\ f^{(2)}_{N_{1}N_{1}}N_{1}^{(1)}N_{1}^{(2)}+2f^{(2)}_{N_{1}N_{2}}\left( N_{1}^{(2)}N_{1}^{(2)}+N_{1}^{(2)}N_{1}^{(2)}\right) +f^{(2)}_{N_{2}N_{2}}N_{2}^{(1)}N_{2}^{(2)} \end{array} \right) \nonumber \\&-\dfrac{1}{6}\left( \begin{array}{lc} f^{(1)}_{N_{1}N_{1}N_{1}}\left( N_{1}^{(1)}\right) ^{3}+3f^{(1)}_{N_{1}N_{1}N_{2}}\left( N_{1}^{(2)}\right) ^{2}N_{2}^{(1)}+3f^{(1)}_{N_{1}N_{2}N_{2}}N_{1}^{(1)}\left( N_{2}^{(1)}\right) ^{2}+f^{(1)}_{N_{2}N_{2}N_{2}}\left( N_{2}^{(1)}\right) ^{3}\\ f^{(2)}_{N_{1}N_{1}N_{1}}\left( N_{1}^{(1)}\right) ^{3}+3f^{(2)}_{N_{1}N_{1}N_{2}}\left( N_{1}^{(2)}\right) ^{2}N_{2}^{(1)}+3f^{(2)}_{N_{1}N_{2}N_{2}}N_{1}^{(1)}\left( N_{2}^{(1)}\right) ^{2}+f^{(2)}_{N_{2}N_{2}N_{2}}\left( N_{2}^{(1)}\right) ^{3} \end{array} \right) \nonumber \\&\triangleq \left( \begin{array}{c} G_{N_{1}}\\ G_{N_{2}} \end{array} \right) . \end{aligned}$$

(A4)

For the first order, after solving (A2, we find

$$\begin{aligned} \left( \begin{array}{c} N_{1}^{(1)}\\ N_{2}^{(1)} \end{array} \right) =\left( \begin{array}{c} \Phi \\ 1 \end{array} \right) \left( \sum _{i=1}^{3}V_{i}\exp (jn_{i}.r)+\right) +c.c., \end{aligned}$$

where $\Phi =\dfrac{\delta _{22}n_{T}^{2}}{C-\delta _{21}^\mathrm{T}n_{T}^{2}}$ and $\vert n_{i}\vert =n_{T}$. $V_{i}$ represents the amplitude of the corresponding $\exp (jn_{i}.r), \quad \text {for} \ i=1,2,3$. Applying the Fredholm solvability conditions [42], the vector function at the right-hand side of Eq. (A3) should be orthogonal to the zero eigenvectors of the adjoint operator $L_{T}^{*}$ of $L_{T}$. Thus, the zero eigenvectors of the operator $L_{T}^{*}$ are

$$\begin{aligned} \left( \begin{array}{c} 1\\ \chi \end{array} \right) \exp (-jn_{i}.r)+c.c.,\quad i=1,2,3, \end{aligned}$$

(A5)

where $\chi =\dfrac{\delta _{11}n_{T}^{2}-A}{C-\delta _{21}^\mathrm{T}n_{T}^{2}}$. Using the orthogonality condition, we find

$$\begin{aligned} (1,\chi )\left( \begin{array}{c} F^{i}_{N_{1}}\\ F^{i}_{N_{2}} \end{array} \right) =0,\quad i=1,2,3, \end{aligned}$$

(A6)

where $F^{(i}_{N_{1}}$ and $F^{i}_{N_{2}}$ stand for coefficients corresponding to $\exp (-jn_{i}.r)$ for $F_{N_{1}}$, $F_{N_{2}}$, respectively.

From (A6), we have

$$\begin{aligned} \left\{ \begin{array}{lcccc} (\Phi +\chi )\dfrac{\partial V_{1}}{\partial t_{1}}=-n^{2}_{T}\delta _{21}^{(1)}\Phi \chi V_{1}+(f_{20}^{(1)}+\chi f_{20}^{(2)})\overline{V_{2}} \ \overline{V_{3}},\\ (\Phi +\chi )\dfrac{\partial V_{2}}{\partial t_{1}}=-n^{2}_{T}\delta _{21}^{(1)}\Phi \chi V_{2}+(f_{20}^{(1)}+\chi f_{20}^{(2)})\overline{V_{1}} \ \overline{V_{3}}, \\ (\Phi +\chi )\dfrac{\partial V_{3}}{\partial t_{1}}=-n^{2}_{T}\delta _{21}^{(1)}\Phi \chi V_{3}+(f_{20}^{(1)}+\chi f_{20}^{(2)})\overline{V_{1}} \ \overline{V_{2}}, \end{array} \right. \end{aligned}$$

(A7)

where

$$\begin{aligned} \left\{ \begin{array}{lcccc} f_{20}^{(1)}=f_{N_{1}N_{1}}^{(1)}\Phi ^{2}+2f_{N_{1}N_{2}}^{(1)}\Phi +f_{N_{2}N_{2}}^{(1)}, \\ f_{20}^{(2)}=f_{N_{1}N_{1}}^{(2)}\Phi ^{2}+2f_{N_{1}N_{2}}^{(2)}\Phi +f_{N_{2}N_{2}}^{(2)}. \end{array} \right. \end{aligned}$$

(A8)

From a similar proceeding in (A3), one can find

$$\begin{aligned} \left( \begin{array}{c} N_{1}^{(2)}\\ N_{2}^{(2)} \end{array} \right)= & {} \left( \begin{array}{c} X_{0}\\ Y_{0} \end{array} \right) +\sum _{i=1}^{3}\left( \begin{array}{c} X_{i}\\ Y_{i} \end{array} \right) \exp (jn_{i}.r)+\sum _{i=1}^{3}\left( \begin{array}{c} X_{ii}\\ Y_{ii} \end{array} \right) \exp (2jn_{i}.r). \nonumber \\&+\sum _{i=1}^{3}\left( \begin{array}{c} X_{12}\\ Y_{12} \end{array} \right) \exp (j(n_{1}-n_{2}).r)+\sum _{i=1}^{3}\left( \begin{array}{c} X_{23}\\ Y_{23} \end{array} \right) \exp (j(n_{2}-n_{3}).r) \nonumber \\&+\sum _{i=1}^{3}\left( \begin{array}{c} X_{31}\\ Y_{31} \end{array} \right) \exp (j(n_{3}-n_{1}).r)+c.c. \end{aligned}$$

(A9)

Clearly, the coefficients in (A9) is determined whether we solve the linear equations of $\exp (0), \ \exp (jn_{i}.r), \ \exp (2jn_{i}.r)$ and $\exp (j(n_{1}-n_{2}).r)$ denoting by

$$\begin{aligned} \Upsilon _{n}=\left( \begin{array}{cc} A-\delta _{11}n^{2} &{}B-\delta _{12}n^{2}\\ C-\delta _{21}n^{2} &{} -\delta _{22}n^{2} \end{array} \right) . \end{aligned}$$

(A10)

Then, for $X_{i}=\Phi Y_{i}, \ i=1,2,3$, we have

$$\begin{aligned} \left( \begin{array}{c} X_{0}\\ Y_{0} \end{array} \right)= & {} -\Upsilon ^{-1}_{0}\left( \begin{array}{c} f^{(1)}_{20}\\ f^{(2)}_{20} \end{array} \right) \left( \vert V_{1} \vert ^{2}+\vert V_{2} \vert ^{2}+\vert V_{3} \vert ^{2}\right) =\left( \begin{array}{c} W_{N_{1}0}\\ W_{N_{2}0} \end{array} \right) \left( \vert V_{1} \vert ^{2}+\vert V_{2} \vert ^{2}+\vert V_{3} \vert ^{2}\right) .\\ \left( \begin{array}{c} X_{11}\\ Y_{11} \end{array} \right)= & {} -\Upsilon ^{-1}_{2n_{T}}\left( \begin{array}{c} \dfrac{f^{(1)}_{20}}{2}\\ \dfrac{f^{(2)}_{20}}{2} \end{array} \right) V_{1}^{2}=\left( \begin{array}{c} W_{N_{1}1}\\ W_{N_{2}1} \end{array} \right) V_{1}^{2} \end{aligned}$$

and

$$\begin{aligned} \left( \begin{array}{c} X_{12}\\ Y_{12} \end{array} \right) =-\Upsilon ^{-1}_{\sqrt{3}n_{T}}\left( \begin{array}{c} f^{(1)}_{20}\\ f^{(2)}_{20} \end{array} \right) V_{1}\overline{V_{2}}=\left( \begin{array}{c} W_{N_{1}2}\\ W_{N_{2}2} \end{array} \right) V_{1}\overline{V_{2}}. \end{aligned}$$

At $O(\epsilon ^{3})$, we have

$$\begin{aligned}&\left( \begin{array}{c} G_{N_{1}}^{1}\\ G_{N_{1}}^{2} \end{array} \right) =\left( \begin{array}{c} \Phi \left( \dfrac{\partial Y_{1}}{\partial t_{1}}+\dfrac{\partial V_{1}}{\partial t_{2}}\right) \\ \dfrac{\partial Y_{1}}{\partial t_{1}}+\dfrac{\partial V_{1}}{\partial t_{2}} \end{array} \right) +n_{T}^{2}J\left( \begin{array}{c} \Phi Y_{1}\\ Y_{1} \end{array} \right) +n_{T}^{2}J\left( \begin{array}{c} \Phi V_{1}\\ V_{1} \end{array} \right) \nonumber \\&-\left( \begin{array}{lcccc} ((f^{(1)}_{N_{1}N_{1}}\Phi +f^{(1)}_{N_{1}N_{2}})(W_{N_{1}0}+W_{N_{1}1})+(f^{(1)}_{N_{1}N_{2}}\Phi +f^{(1)}_{N_{2}N_{2}})(Z_{N_{2}}0+W_{N_{2}1}))\vert V_{1}\vert ^{2}\\ +\left[ (f^{(1)}_{N_{1}N_{1}}\Phi {+}f^{(1)}_{N_{1}N_{2}})(W_{N_{1}0}{+}W_{N_{1}2}){+}(f^{(1)}_{N_{1}N_{2}}\Phi {+}f^{(1)}_{N_{2}N_{2}})(W_{N_{2}0}{+}W_{N_{2}2})(\vert V_{2}\vert ^{2}{+}\vert V_{3}\vert ^{2})\right] V_{1}{+}f^{(1)}_{20}\left( \overline{V_{2}} \ \overline{Y_{3}}{+}\overline{V_{3}} \ \overline{Y_{2}}\right) \\ ((f^{(2)}_{N_{1}N_{1}}\Phi +f^{(2)}_{N_{1}N_{2}})(W_{N_{1}0}+W_{N_{1}1})+(f^{(2)}_{N_{1}N_{2}}\Phi +f^{(2)}_{N_{2}N_{2}})(Z_{N_{2}}0+W_{N_{2}1}))\vert V_{1}\vert ^{2}\\ +\left[ (f^{(2)}_{N_{1}N_{1}}\Phi {+}f^{(2)}_{N_{1}N_{2}})(W_{N_{1}0}{+}W_{N_{1}2}){+}(f^{(2)}_{N_{1}N_{2}}\Phi {+}f^{(2)}_{N_{2}N_{2}})(W_{N_{2}0}{+}W_{N_{2}2})(\vert V_{2}\vert ^{2}{+}\vert V_{3}\vert ^{2})\right] V_{1}{+}f^{(1)}_{20}\left( \overline{V_{2}} \ \overline{Y_{3}}{+}\overline{V_{3}} \ \overline{Y_{2}}\right) \end{array} \right) \nonumber \\&-\left( \begin{array}{c} \left( \vert V_{1}\vert ^{2}+\vert V_{2}\vert ^{2}+\vert V_{3}\vert ^{2}\right) \left( f^{(1)}_{N_{1}N_{1}N_{1}}\Phi ^{3}+3f^{(1)}_{N_{1}N_{2}N_{2}}\Phi +f^{(1)}_{N_{2}N_{2}N_{2}}\right) \\ \left( \vert V_{1}\vert ^{2}+\vert V_{2}\vert ^{2}+\vert V_{3}\vert ^{2}\right) \left( f^{(2)}_{N_{1}N_{1}N_{1}}\Phi ^{3}+3f^{(2)}_{N_{1}N_{2}N_{2}}\Phi +f^{(2)}_{N_{2}N_{2}N_{2}}\right) . \end{array} \right) \end{aligned}$$

(A11)

Similarly, from the permutation of the parameters of V and Y, we can obtain the other coefficients $(G_{N_{1}}^{2},G_{N_{2}}^{2})^\mathrm{T}$ and $(G_{N_{1}}^{3},G_{N_{2}}^{32})^\mathrm{T}$. Now, by using Fredholm solvability condition

$$\begin{aligned} (1,\chi )\left( \begin{array}{c} G^{i}_{N_{1}}\\ G^{i}_{N_{2}} \end{array} \right) =0,\quad i=1,2,3, \end{aligned}$$

it follows that

$$\begin{aligned} \left\{ \begin{array}{lcccc} (\Phi +\chi )\left( \dfrac{\partial V_{1}}{\partial t_{2}}+\dfrac{\partial Y_{1}}{\partial t_{1}}\right) =-n^{2}_{T}\Phi \chi \left( \delta ^{1}_{21}Y_{1}+\delta ^{2}_{21}V_{1}\right) ,\\ \qquad \qquad \qquad \qquad \quad \ \ \ +\Lambda _{1}\left( \overline{V_{2}} \ \overline{Y_{3}}+\overline{V_{3}} \ \overline{Y_{2}}\right) +\left( G_{1}\vert V_{1} \vert ^{2}+G_{2}\left( \vert V_{2} \vert ^{2}+\vert V_{3} \vert ^{2}\right) \right) V_{1},\\ (\Phi +\chi )\left( \dfrac{\partial V_{2}}{\partial t_{2}}+\dfrac{\partial Y_{2}}{\partial t_{1}}\right) =-n^{2}_{T}\Phi \chi \left( \delta ^{1}_{21}Y_{2}+\delta ^{2}_{21}V_{2}\right) ,\\ \qquad \qquad \qquad \qquad \quad \ \ \ +\Lambda _{1}\left( \overline{V_{3}} \ \overline{Y_{1}}+\overline{V_{1}} \ \overline{Y_{3}}\right) +\left( G_{1}\vert V_{2} \vert ^{2}+G_{2}\left( \vert V_{1} \vert ^{2}+\vert V_{3} \vert ^{2}\right) \right) V_{2}, \\ (\Phi +\chi )\left( \dfrac{\partial V_{3}}{\partial t_{2}}+\dfrac{\partial Y_{3}}{\partial t_{1}}\right) =-n^{2}_{T}\Phi \chi \left( \delta ^{1}_{21}Y_{3}+\delta ^{2}_{21}V_{3}\right) ,\\ \qquad \qquad \qquad \qquad \quad \ \ \ +\Lambda _{1}\left( \overline{V_{1}} \ \overline{Y_{2}}+\overline{V_{2}} \ \overline{Y_{1}}\right) +\left( G_{1}\vert V_{3} \vert ^{2}+G_{2}\left( \vert V_{1} \vert ^{2}+\vert V_{2} \vert ^{2}\right) \right) V_{3} \end{array} \right. \end{aligned}$$

(A12)

with

$$\begin{aligned} \begin{array}{lcccc} \Lambda _{1}=f_{20}^{(1)}+\chi f_{20}^{(2)},\\ G_{1}=-\left( f_{N_{1}N_{1}}^{(1)}\Phi +f_{N_{1}N_{2}}^{(1)}+\chi \left( f_{N_{1}N_{1}}^{(2)}\Phi +f_{N_{1}N_{2}}^{(2)}\right) \right) \left( W_{N_{1}0}+W_{N_{1}1}\right) ,\\ \qquad \ -\left( f_{N_{1}N_{2}}^{(1)}\Phi +f_{N_{2}N_{2}}^{(1)}+\chi \left( f_{N_{1}N_{2}}^{(2)}\Phi +f_{N_{2}N_{2}}^{(2)}\right) \right) \left( W_{N_{2}0}+W_{N_{2}1}\right) -f_{30}^{(1)}-\chi f_{30}^{(2)},\\ G_{2}=-\left( f_{N_{1}N_{1}}^{(1)}\Phi +f_{N_{1}N_{2}}^{(1)}+\chi \left( f_{N_{1}N_{1}}^{(2)}\Phi +f_{N_{1}N_{2}}^{(2)}\right) \right) \left( W_{N_{1}0}+W_{N_{1}1}\right) , \\ \qquad \ -\left( f_{N_{1}N_{2}}^{(1)}\Phi +f_{N_{2}N_{2}}^{(1)}+\chi \left( f_{N_{1}N_{2}}^{(2)}\Phi +f_{N_{2}N_{2}}^{(2)}\right) \right) \left( W_{N_{2}0}+W_{N_{2}1}\right) -f_{30}^{(1)}-\chi f_{30}^{(2)},\\ f_{30}^{(1)}=f_{N_{1}N_{1}N_{1}}^{(1)}\Phi ^{3}+3f_{N_{1}N_{1}N_{2}}^{(1)}\Phi ^{2}+3f_{N_{1}N_{2}N_{2}}^{(1)}\Phi +f_{N_{2}N_{2}N_{2}}^{(1)},\\ f_{30}^{(2)}=f_{N_{1}N_{1}N_{1}}^{(2)}\Phi ^{3}+3f_{N_{1}N_{1}N_{2}}^{(2)}\Phi ^{2}+3f_{N_{1}N_{2}N_{2}}^{(2)}\Phi +f_{N_{2}N_{2}N_{2}}^{(2)}. \end{array} \end{aligned}$$

(A13)

The amplitude $M_{i}$ can be widened as

$$\begin{aligned} M_{i}=\epsilon V_{i}+\epsilon ^{2}Y_{i}+O(\epsilon ^{2}). \end{aligned}$$

Combining the above equation with (3.9), we obtain the following amplitude equation

$$\begin{aligned} \left\{ \begin{array}{lccc} \tau _{0}\dfrac{\partial M_{1}}{\partial t}=\mu M_{1}+\Lambda \overline{M_{2}} \ \overline{M_{3}}-\left( \kappa _{1}\vert M_{1} \vert ^{2}+\kappa _{2}\vert M_{2} \vert ^{2}+\vert M_{3} \vert ^{2}\right) M_{1},\\ \tau _{0}\dfrac{\partial M_{2}}{\partial t}=\mu M_{2}+\Lambda \overline{M_{1}} \ \overline{M_{3}}-\left( \kappa _{1}\vert M_{2} \vert ^{2}+\kappa _{2}\vert M_{3} \vert ^{2}+\vert M_{1} \vert ^{2}\right) M_{2}, \\ \tau _{0}\dfrac{\partial M_{3}}{\partial t}=\mu M_{3}+\Lambda \overline{M_{1}} \ \overline{M_{2}}-\left( \kappa _{1}\vert M_{2} \vert ^{2}+\kappa _{2}\vert M_{3} \vert ^{2}+\vert M_{1} \vert ^{2}\right) M_{3}, \end{array} \right. \end{aligned}$$

(A14)

where

$$\begin{aligned} \tau _{0}=-\dfrac{(\Phi +\chi )}{n^{2}_{T}\Phi \chi \delta _{21}^\mathrm{T}},\quad \mu =\dfrac{\delta _{21}-\delta _{21}^\mathrm{T}}{\delta _{21}^\mathrm{T}},\quad \Lambda =-\dfrac{\Lambda _{1}}{n^{2}_{T}\Phi \chi \delta _{21}^\mathrm{T}},\\\quad \kappa _{1}=-\dfrac{G_{1}}{n^{2}_{T}\Phi \chi \delta _{21}^\mathrm{T}},\quad \kappa _{2}=-\dfrac{G_{2}}{n^{2}_{T}\Phi \chi \delta _{21}^\mathrm{T}}. \end{aligned}$$