Modeling Refuge Effect of Submerged Macrophytes in Lake System

Abstract

This paper considers a significant problem in biological control of algae issue in ecological environment. A four-dimensional dynamic model is carefully formulated to characterize the interactions among phytoplankton, submerged macrophyte, zooplankton, and general fish class in a lake ecosystem. The predation relationship is modeled by Beddington–DeAngelis functional responses derived from the classical Holling time budget arguments. Qualitative analyses of the global dynamics show that the system can generate very rich dynamics with potentially 10 different equilibria and several bistable scenarios. We perform analysis on the existence and local stability of equilibria and explore the refuge effect of macrophyte on the zooplankton with numerical simulations on aquatic ecosystems. We also discuss effective methods of biological control used to restrain the increase of phytoplankton. Our study shows the proposed model could have rich and complex dynamics including but not limited to bistable and chaotic phenomenon. Numerical simulation results demonstrate that both the refuge constant and the density of the macrophytes are two key factors where refuge effects take place. In addition, the intraspecific competition between the macrophyte and the phytoplankton can also affect the macrophyte’s refuge effect. Our analytical and simulation results suggest that macrophytes provide structure and shelter against predation for zooplankton such that it could restore the zooplankton population, and that planting macrophyte properly might achieve the purpose of controlling algae growth.

Appendices

Appendix 1: Proof of Theorem 2: Existence and Stability of $E_{pm}$, $E_{pz}$, and $E_{pf}$

Note that

$$\begin{aligned} P^{*}_{4}=\displaystyle \frac{K_{1}r_{2}(K_{2}\eta _{1}-r_{1})}{\eta _{1}\eta _{2}K_{1}K_{2}-r_{1}r_{2}},~~ M^{*}_{4}=\displaystyle \frac{r_{1}K_{2}(K_{1}\eta _{2}-r_{2})}{\eta _{1}\eta _{2}K_{1}K_{2}-r_{1}r_{2}}; \end{aligned}$$

we have

$$\begin{aligned} J(E_{pm})= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} a_{11} &{} a_{12} &{} a_{13}&{} a_{14}\\ a_{21} &{} a_{22} &{} 0 &{} 0\\ 0 &{} 0 &{} a_{33} &{} 0\\ 0 &{} 0 &{} 0 &{} a_{44} \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} a_{11}&=-\displaystyle \frac{r_{1}P^{*}_{4}}{K_{1}},&a_{12}&=-\eta _{1}P^{*}_{4},&a_{13}&=-\alpha _{1}P^{*}_{4},&a_{14}&=-\displaystyle \frac{a_{1}P^{*}_{4}}{1+\alpha P^{*}_{4}+\beta M^{*}_{4}},\\ a_{21}&\!=\!-\eta _{2}M^{*}_{4},&a_{22}&\!=\!-\displaystyle \frac{r_{2}M^{*}_{4}}{K_{2}},&a_{33}&\!=\!\alpha _{2}P^{*}_{4}-d_{z},&a_{44}&\!=\!\displaystyle \frac{b_{1}P^{*}_{4}}{1\!+\!\alpha P^{*}_{4}\!+\!\beta M^{*}_{4}}-d_{f}. \end{aligned}$$

It is clear that $a_{33}$ and $a_{44}$ are two of the eigenvalues for $J(E_{pm})$ and the other two eigenvalues solve the following equation

$$\begin{aligned} \lambda ^{2}+c_{1}\lambda +c_{2}=0, \end{aligned}$$

where

$$\begin{aligned} c_{1}=\displaystyle \frac{r_{1}P^{*}_{4}}{K_{1}}+\displaystyle \frac{r_{2}M^{*}_{4}}{K_{2}}>0,~~ c_{2}=\displaystyle \frac{r_{1}r_{2}-\eta _{1}\eta _{2}K_{1}K_{2}}{K_{1}K_{2}}P^{*}_{4}M^{*}_{4}>0. \end{aligned}$$

It is not difficult to show that all the eigenvalues of $J(E_{pm})$ have negative real parts. Therefore, $E_{pm}$ is locally asymptotically stable.

For $E_{pz}$, we have

$$\begin{aligned} J(E_{pz})= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} -\displaystyle \frac{r_{1}P^{*}_{5}}{K_{1}} &{} -\eta _{1}P^{*}_{5} &{} -\alpha _{1}P^{*}_{5} &{} -\displaystyle \frac{a_{1}P^{*}_{5}}{1+\alpha P^{*}_{5}+\gamma Z^{*}_{5}}\\ 0 &{} r_{2}-\eta _{2}P^{*}_{5} &{} 0 &{} 0\\ \alpha _{2}Z^{*}_{5} &{} 0 &{} 0 &{} -\displaystyle \frac{a_{2}Z^{*}_{5}}{1+\alpha P^{*}_{5}+\gamma Z^{*}_{5}}\\ 0 &{} 0 &{} 0 &{} \displaystyle \frac{b_{1}P^{*}_{5}+b_{2}Z^{*}_{5}}{1+\alpha P^{*}_{5}+\gamma Z^{*}_{5}}-d_{f} \end{array}\right) \end{aligned}$$

whose two eigenvalues are

$$\begin{aligned} r_{2}-\eta _{2}P^{*}_{5},~~\displaystyle \frac{b_{1}P^{*}_{5}+b_{2}Z^{*}_{5}}{1+\alpha P^{*}_{5}+\gamma Z^{*}_{5}}-d_{f} \end{aligned}$$

and other two eigenvalues solve $\lambda ^{2}+c_{1}\lambda +c_{2}=0$ with $c_{1}=r_{1}P^{*}_{5}/K_{1}>0$ and $c_{2}=\alpha _{1}\alpha _{2}P^{*}_{5}Z^{*}_{5}.$ Direct calculation shows that all the eigenvalues of $J(E_{pz})$ have negative real parts. Therefore, $E_{pz}$ is locally asymptotically stable.

Direct calculation leads to the first claim on the existence of $E_{pf}$. In addition,

$$\begin{aligned} J(E_{pf})=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} -\displaystyle \frac{r_{1}P^{*}_{6}}{K_{1}}+\displaystyle \frac{a_{1}\alpha P^{*}_{6}F^{*}_{6}}{(1+\alpha P^{*}_{6})^{2}} &{} -\eta _{1}P^{*}_{6}+\displaystyle \frac{a_{1}\beta F^{*}_{6}P^{*}_{6}}{(1+\alpha P^{*}_{6})^{2}} &{} -\alpha _{1}P^{*}_{6}+\displaystyle \frac{a_{1}\gamma P^{*}_{6}F^{*}_{6}}{(1+\alpha P^{*}_{6})^{2}} &{} -\displaystyle \frac{a_{1}P^{*}_{6}}{1+\alpha P^{*}_{6}} \\ 0 &{} r_{2}-\eta _{2}P^{*}_{6} &{} 0 &{} 0\\ 0 &{} 0 &{} \alpha _{2}P^{*}_{6}-\displaystyle \frac{a_{2}F^{*}_{6}}{1+\alpha P^{*}_{6}}-d_{z} &{} 0\\ \displaystyle \frac{b_{1}F^{*}_{6}}{(1+\alpha P^{*}_{6})^{2}} &{} -\displaystyle \frac{b_{1}\beta P^{*}_{6}F^{*}_{6}}{(1+\alpha P^{*}_{6})^{2}} &{} \displaystyle \frac{b_{2}F^{*}_{6}(1+\alpha P^{*}_{6})- b_{1}\gamma P^{*}_{6}F^{*}_{6}}{(1+\alpha P^{*}_{6})^{2}}&{} 0 \end{array}\right) . \end{aligned}$$

Two of the eigenvalues of $J(E_{pf})$ are

$$\begin{aligned} r_{2}-\eta _{2}P^{*}_{6},~~-d_{z}-\displaystyle \frac{a_{2}r_{1}}{a_{1}}+\displaystyle \frac{K_{1}\alpha _{2}a_{1}+a_{2}r_{1}}{a_{1}K_{1}}P^{*}_{6} \end{aligned}$$

and the other two eigenvalues are given by the following equation

$$\begin{aligned} \lambda ^{2}+c_{1}\lambda +c_{2}=0, \end{aligned}$$

where

$$\begin{aligned} c_{1}=\displaystyle \frac{r_{1}P_{6}^{*}(1+2\alpha P_{6}^{*}-\alpha K_{1})}{K_{1}(1+\alpha P_{6}^{*})}>0,~~c_{2}=\displaystyle \frac{a_{1}b_{1}P^{*}_{6}F^{*}_{6}}{(1+\alpha P^{*}_{6})^{3}}>0. \end{aligned}$$

Since all the eigenvalues of $J(E_{pf})$ have negative real parts, $E_{pf}$ is locally asymptotically stable. $\square$

Appendix 2: Proof of Theorem 3: Existence and Stability of $E_{pmz}$

From (7), it follows that

$$\begin{aligned} P^{*}_{7}\!=\!\displaystyle \frac{d_{z}}{\alpha _{2}},~~M^{*}_{7}\!=\!\displaystyle \frac{K_{2}(r_{2}\alpha _{2}\!-\!\eta _{2}d_{z})}{r_{2}\alpha _{2}},~~ Z^{*}_{7}\!=\!\displaystyle \frac{r_{1}r_{2}(K_{1}\alpha _{2}-d_{z})\!-\!\eta _{1}K_{1}K_{2}(r_{2}\alpha _{2}\!-\!\eta _{2}d_{z})}{K_{1}r_{2}\alpha _{1}\alpha _{2}}. \end{aligned}$$

Then the existence of $E_{pmz}$ is obvious. In addition,

$$\begin{aligned} J(E_{pmz})=\left( \begin{array}{cccc} -\displaystyle \frac{r_{1}P^{*}_{7}}{K_{1}} &{} -\eta _{1}P^{*}_{7} &{} -\alpha _{1}P^{*}_{7} &{} -\displaystyle \frac{a_{1}P^{*}_{7}}{1+\alpha P^{*}_{7}+\beta M^{*}_{7}+\gamma Z^{*}_{7}} \\ -\eta _{2}M^{*}_{7} &{} -\displaystyle \frac{r_{2}M^{*}_{7}}{K_{2}} &{} 0 &{} 0\\ \alpha _{2}Z^{*}_{7} &{} 0 &{} 0 &{} -\displaystyle \frac{a_{2}Z^{*}_{7}}{1+\alpha P^{*}_{7}+\beta M^{*}_{7}+\gamma Z^{*}_{7}}\\ 0 &{} 0 &{} 0&{} \displaystyle \frac{b_{1} P^{*}_{7}+b_{2}Z^{*}_{7}}{1+\alpha P^{*}_{7}+\beta M^{*}_{7}+\gamma Z^{*}_{7}}-d_{f} \end{array}\right) . \end{aligned}$$

Then, $(b_{1} P^{*}_{7}+b_{2}Z^{*}_{7})/(1+\alpha P^{*}_{7}+\beta M^{*}_{7}+\gamma Z^{*}_{7})-d_{f}$ is one of the eigenvalues of $J(E_{pmz})$ and the other three eigenvalues solve

$$\begin{aligned} \lambda ^{3}+c_{1}\lambda ^{2}+c_{2}\lambda +c_{3}=0, \end{aligned}$$

where

$$\begin{aligned} c_{1}= & {} \displaystyle \frac{r_{1}P^{*}_{7}}{K_{1}}+\displaystyle \frac{r_{2}M^{*}_{7}}{K_{2}}>0,~~c_{3}=\displaystyle \frac{\alpha _{1}\alpha _{2}r_{2}P^{*}_{7}M^{*}_{7}Z^{*}_{7}}{K_{2}}>0,\nonumber \\ c_{2}= & {} \displaystyle \frac{(r_{1}r_{2}-\eta _{1}\eta _{2}K_{1}K_{2})P^{*}_{7}M^{*}_{7}+\alpha _{1}\alpha _{2}K_{1}K_{2}P_{7}^{*}Z_{7}^{*}}{K_{1}K_{2}},\nonumber \\ c_{1}c_{2}-c_{3}\!= & {} \!\displaystyle \frac{(r_{1}K_{2}P^{*}_{7}\!+\!K_{1}r_{2}M^{*}_{7})(r_{1}r_{2}\!-\!\eta _{1}\eta _{2}K_{1}K_{2})P^{*}_{7}M^{*}_{7}\!+\!\alpha _{1}\alpha _{2}r_{1}K_{1}K^{2}_{2}P^{*^{2}}_{7}Z^{*}_{7}}{K^{2}_{1}K^{2}_{2}}>0. \end{aligned}$$

By Routh–Hurwitz criteria, all the eigenvalues of $J(E_{pmz})$ have negative real parts. Therefore, $E_{pmz}$ is locally asymptotically stable. $\square$

Appendix 3: Proof of Theorem 4: Existence and Stability of $E_{pmf}$

Note that $P^{*}_{8}, M^{*}_{8}, F^{*}_{8}$ solve (8) and (8) is equivalent to

$$\begin{aligned} M=-\displaystyle \frac{\eta _{2}K_{2}}{r_{2}}P+K_{2},~~M=\displaystyle \frac{b_{1}-\alpha d_{f}}{\beta d_{f}}P-\displaystyle \frac{1}{\beta },~~F=\displaystyle \frac{b_{1}\left[ r_{1}\left( 1-\displaystyle \frac{P}{K_{1}}\right) -\eta _{1}M\right] P}{a_{1}d_{f}}; \end{aligned}$$

(19)

then, $(P^{*}_{8}, M^{*}_{8})$ is the intersection of the straight lines $l_1$ and $l_2$, which are defined by the first two equations of (19), respectively, and

$$\begin{aligned} F_8^*=\displaystyle \frac{b_{1}\left[ \displaystyle \frac{\eta _{1}\eta _{2}K_{1}K_{2} -r_{1}r_{2}}{K_{1}r_{2}}P^{2}-(\eta _{1}K_{2}-r_{1})P\right] }{a_{1}d_{f}}|_{P=P_8^*}. \end{aligned}$$

(20)

In fact, if

$$\begin{aligned} 0<\displaystyle \frac{d_{f}}{b_{1}-\alpha d_{f}}<\min \left\{ K_{1},~~\displaystyle \frac{r_{2}}{\eta _{2}}\right\} , \end{aligned}$$

(21)

then $l_1$ and $l_2$ have a unique intersection in the positive quadrant of $P-M$ plane (see Fig 10) with

$$\begin{aligned} P_{8}^{*}=\displaystyle \frac{(K_{2}\beta +1)d_{f}r_{2}}{(b_{1}-\alpha d_{f})r_{2}+\eta _{2}K_{2}\beta d_{f}}. \end{aligned}$$

(22)

According to the characteristics of $l_3$ and the relative location of $l_3$ and $l_4$, if one of (9)–(12) is satisfied, then the parabola $l_3$ defined by (20) and the straight line $l_4$ defined by (22) intersect at a unique point in the positive quadrant of $P-F$ plane (see Fig. 11 for more details), which implies that $F_8^*$ is positive.

Fig. 10

Location and intersection of the straight lines $l_1$ and $l_2$. $l_1$ and $l_2$ have a unique intersection in the positive quadrant if (21) is satisfied

Fig. 11

Location and intersection of the parabola defined by (20) and the straight line $l_4$ defined by (22), which imply that $F_8^*$ is positive. a (9) is satisfied. b (10) is satisfied. c (11) or (12) is satisfied. d $\eta _{1}\eta _{2}K_{1}K_{2}-r_{1}r_{2}=0$ ($l_3$ degenerates to a straight line). Here $A=([(\eta _{1}K_{2}-r_{1})K_{1}r_{2}]/[\eta _{1}\eta _{2}K_{1}K_{2}-r_{1}r_{2}],0)$, $B=([(K_{2}\beta +1)d_{f}r_{2}]/[(b_{1}-\alpha d_{f})r_{2}+\eta _{2}K_{2}\beta d_{f}],0)$

Due to the different cases for the existence of $E_{pmf}$, it seems tedious to derive sufficient conditions for the stability of $E_{pmf}$. As an example, we only deal with one case. The other cases can be explored similarly. For the stability of $E_{pmf}$, consider the Jacobian at $E_{pmf}$

$$\begin{aligned} J(E_{pmf})= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} a_{11} &{} a_{12} &{} a_{13}&{} a_{14}\\ a_{21} &{} a_{22} &{} 0 &{} 0\\ 0 &{} 0 &{} a_{33} &{} 0\\ a_{41} &{} a_{42} &{} a_{43}&{} 0 \end{array}\right) , \end{aligned}$$

where

$$\begin{aligned} a_{11}&=-\displaystyle \frac{r_{1}P^{*}_{8}}{K_{1}}+\displaystyle \frac{a_{1}\alpha P^{*}_{8}F^{*}_{8}}{(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{2}},&a_{12}&=-\eta _{1}P^{*}_{8}+\displaystyle \frac{a_{1}\beta P^{*}_{8}F^{*}_{8}}{(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{2}},\\ a_{13}&=-\alpha _{1}P^{*}_{8}+\displaystyle \frac{a_{1}\gamma P^{*}_{8}F^{*}_{8}}{(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{2}},&a_{14}&=-\displaystyle \frac{a_{1}P^{*}_{8}}{1+\alpha P^{*}_{8}+\beta M^{*}_{8}},\\ a_{21}&=-\eta _{2}M^{*}_{8},&a_{22}&=-\displaystyle \frac{r_{2}M^{*}_{8}}{K_{2}},\\ a_{33}&=\alpha _{2}P^{*}_{8}-d_{z}-\displaystyle \frac{a_{2} \left[ r_{1}\left( 1-\displaystyle \frac{P^{*}_{8}}{K_{1}}\right) -\eta _{1}M^{*}_{8}\right] }{a_{1}},&a_{41}&=\displaystyle \frac{b_{1}F^{*}_{8}(1+\beta M^{*}_{8})}{(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{2}},\\ a_{42}&=-\displaystyle \frac{\beta d_{f}F^{*}_{8}}{1+\alpha P^{*}_{8}+\beta M^{*}_{8}},&a_{43}&=\displaystyle \frac{b_{2}F^{*}_{8}(1+\alpha P^{*}_{8}+\beta M^{*}_{8})-b_{1}\gamma P^{*}_{8}F^{*}_{8}}{(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{2}}.\\ \end{aligned}$$

One eigenvalue of $J(E_{pmf})$ is

$$\begin{aligned} a_{33}=\displaystyle \frac{a_{1}K_{1}r_{2}\alpha _{2}+a_{2}r_{1}r_{2}-\eta _{1}\eta _{2}a_{2}K_{1}K_{2}}{a_{1}K_{1}r_{2}}P_{8}^{*}+\displaystyle \frac{a_{2}\eta _{1}K_{2}-a_{2}r_{1}-a_{1}d_{z}}{a_{1}}<0 \end{aligned}$$

and the other eigenvalues solve

$$\begin{aligned} |\lambda E-A|=\lambda ^{3}+c_{1}\lambda ^{2}+c_{2}\lambda +c_{3}, \end{aligned}$$

where

$$\begin{aligned} A= & {} \left( \begin{array}{c@{\quad }c@{\quad }c} a_{11} &{} a_{12} &{} a_{14}\\ a_{21} &{} a_{22} &{} 0 \\ a_{41} &{} a_{42} &{} 0 \end{array}\right) ,\\ c_{1}= & {} \displaystyle \frac{r_{1}r_{2}(b_{1}+d_{f}\alpha )-K_{1}\eta _{2}(b_{1}r_{2} +\alpha d_{f}\eta _{1}K_{2})}{b_{1}K_{1}r_{2}}P^{*}_{8}\\&\quad +\,\displaystyle \frac{b_{1}r_{2}+\alpha d_{f}\eta _{1}K_{2}-\alpha d_{f}r_{1}}{b_{1}}>0,\\ c_{2}= & {} \displaystyle \frac{a_{1}b_{1}P^{*}_{8}F^{*}_{8}(1+\beta M^{*}_{8})}{(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{3}} +\displaystyle \frac{( K_{2}a_{1}\beta \eta _{2}-a_{1}\alpha r_{2})P^{*}_{8}M^{*}_{8} F^{*}_{8}}{K_{2}(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{2}}\\&\quad +\,\displaystyle \frac{(r_{1}r_{2}-\eta _{1}\eta _{2}K_{1}K_{2})P^{*}_{8}M^{*}_{8}}{K_{1}K_{2}}>0,\\ c_{3}= & {} \displaystyle \frac{a_{1}b_{1}r_{2}(1+\beta M^{*}_{8})P^{*}_{8}M^{*}_{8} F^{*}_{8}}{K_{2}(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{3}} +\displaystyle \frac{a_{1}b_{1}\eta _{2}\beta P^{*^{2}}_{8}M^{*}_{8}F^{*}_{8}}{(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{3}}\\&\quad +\displaystyle \frac{(r_{1}r_{2}-\eta _{1}\eta _{2}K_{1}K_{2})P^{*^{2}}_{8}M^{*}_{8}F^{*}_{8}}{K_{1}K_{2}(1+\alpha P^{*}_{8}+\beta M^{*}_{8})^{2}}>0,\\ c_{1}c_{2}-c_{3}= & {} \displaystyle \frac{a_{1}\eta _{2}\beta [a_{1}\alpha ^{2}(\eta _{1} \eta _{2}K_{1}K_{2}-r_{1}r_{2})P_{8}^{*} +K_{1}a_{1}\alpha ^{2}r_{2}(r_{1}-\eta _{1}K_{2})-b_{1}K_{1}] P^{*^{2}}_{8}M^{*}_{8}F^{*}_{8}}{K_{1}(1+\alpha P^{*}_{8} +\beta M^{*}_{8})^{3}}>0. \end{aligned}$$

By Routh–Hurwitz criteria, all the eigenvalues are negative, and hence, $E_{pmf}$ is locally asymptotically stable. $\square$