Impact of awareness on environmental toxins affecting plankton dynamics: a mathematical implication

Abstract

The widespread problem of water pollution due to enhanced concentration of anthropogenic effluents is becoming a global issue. Public environmental awareness may be a plausible factor for the control of toxicants in the aquatic medium. The present paper is devoted to study the impact of awareness among human on reduction of environmental toxins affecting planktonic system. The provision of awareness among people helps to maintain the ecological balance of the system by reducing the input rate of toxicants through anthropogenic sources. The conditions for existence and local asymptotic stability of all feasible steady states of the system are derived. Our study reveals that the system is stable for low or high input rate of toxicants, but for intermediate ranges, the system produces oscillations by destroying the stable dynamics. Moreover, for very large level of pollutants, zooplankton disappears from the system. Importantly, the limited supply of additional food to zooplankton prevents the crash of aquatic food web system. Sensitivity results evoke that environmental toxins can be reduced to a low level by imparting awareness among human, thereby maintaining the rhythm of the planktonic ecosystem.

Appendices

Appendix A

$$\begin{aligned}&\widetilde{N}_1=\widetilde{N}_{11}+\widetilde{N}_{33}+\widetilde{N}_{44},\\&\widetilde{N}_2=\widetilde{N}_{33}\widetilde{N}_{44}+\widetilde{N}_{34}\widetilde{N}_{43}-\widetilde{N}_{23}\widetilde{N}_{31} +\widetilde{N}_{11}(\widetilde{N}_{33}+\widetilde{N}_{44})+\widetilde{N}_{12}\widetilde{N}_{21}-\widetilde{N}_{13}\widetilde{N}_{31},\\&\widetilde{N}_3=\widetilde{N}_{11}(\widetilde{N}_{33}\widetilde{N}_{44}+\widetilde{N}_{34}\widetilde{N}_{43}-\widetilde{N}_{23}\widetilde{N}_{31}) -\widetilde{N}_{23}\widetilde{N}_{31}\widetilde{N}_{44}+\widetilde{N}_{12}\widetilde{N}_{21}(\widetilde{N}_{33}+\widetilde{N}_{44})\\&\quad +\widetilde{N}_{12}\widetilde{N}_{23}\widetilde{N}_{31} -\widetilde{N}_{13}\widetilde{N}_{31}(\widetilde{N}_{21}+\widetilde{N}_{44}),\\&\widetilde{N}_4=\widetilde{N}_{12}\{\widetilde{N}_{21}(\widetilde{N}_{33}\widetilde{N}_{44}+\widetilde{N}_{34}\widetilde{N}_{43})+\widetilde{N}_{23}\widetilde{N}_{31} \widetilde{N}_{44}\} -\widetilde{N}_{32}\widetilde{N}_{44}(\widetilde{N}_{11}\widetilde{N}_{23}+\widetilde{N}_{13}\widetilde{N}_{21}) \end{aligned}$$

with

$$\begin{aligned}&\widetilde{N}_{11}=\frac{\beta a\widetilde{Z}^*}{(a+\widetilde{P}^*)^2}-\frac{r_p}{(1+\gamma c_p\widetilde{P}^*\widetilde{E}^*)^2}\left( 1-\frac{\widetilde{P}^*(2+\gamma c_p\widetilde{P}^*\widetilde{E}^*)}{K_p}\right) , \\&\widetilde{N}_{12}=\frac{\beta \widetilde{P}^*}{a+\widetilde{P}^*},\\&\widetilde{N}_{13}=\frac{r_p\gamma c_p(\widetilde{P}^*)^2}{(1+\gamma c_p\widetilde{P}^*\widetilde{E}^*)^2}\left( 1-\frac{\widetilde{P}^*}{K_p}\right) , \ \widetilde{N}_{21}=\frac{a\beta c_z \widetilde{Z}^*}{(a+\widetilde{P}^*)^2}, \ \widetilde{N}_{23}=\gamma c_e\widetilde{Z}^*, \\&\widetilde{N}_{31}=\widetilde{N}_{32}=\gamma \widetilde{E}^*,\\&\widetilde{N}_{33}=d_e+\gamma (\widetilde{P}^*+\widetilde{Z}^*), \ \widetilde{N}_{34}=\frac{r_ebp}{(b+\widetilde{M}^*)^2}, \ \widetilde{N}_{43}=\frac{\phi }{(1+c\widetilde{E}^*)^2}, \ \widetilde{N}_{44}=\phi _0. \end{aligned}$$

Appendix B

$$\begin{aligned}&\widetilde{A}_1=\widetilde{A}_{11}+\widetilde{A}_{22}+\widetilde{A}_{33}+\widetilde{A}_{44},\\&\widetilde{A}_2=\widetilde{A}_{11}(\widetilde{A}_{22}+\widetilde{A}_{33}+\widetilde{A}_{44})+\widetilde{A}_{22}(\widetilde{A}_{33}+\widetilde{A}_{44}) +\widetilde{A}_{33}\widetilde{A}_{44}+\widetilde{A}_{34}\widetilde{A}_{43}\\&\quad +\widetilde{A}_{12}\widetilde{A}_{21}-\widetilde{A}_{23}\widetilde{A}_{32}-\widetilde{A}_{13}\widetilde{A}_{31},\\&\widetilde{A}_3=\widetilde{A}_{22}(\widetilde{A}_{33}\widetilde{A}_{44}+\widetilde{A}_{34}\widetilde{A}_{43})-\widetilde{A}_{23}\widetilde{A}_{32}\widetilde{A}_{44} +\widetilde{A}_{11}\{\widetilde{A}_{22}(\widetilde{A}_{33}+\widetilde{A}_{44})\\&\quad +\widetilde{A}_{33}\widetilde{A}_{44}+\widetilde{A}_{34}\widetilde{A}_{43} -\widetilde{A}_{23}\widetilde{A}_{32}\}\\&\quad +\widetilde{A}_{12}\{\widetilde{A}_{21}(\widetilde{A}_{33}+\widetilde{A}_{44})+\widetilde{A}_{23}\widetilde{A}_{31}\} -\widetilde{A}_{13}\{\widetilde{A}_{31}(\widetilde{A}_{22}+\widetilde{A}_{44})+\widetilde{A}_{21}\widetilde{A}_{32}\},\\&\widetilde{A}_4=\widetilde{A}_{11}\{\widetilde{A}_{22}(\widetilde{A}_{33}\widetilde{A}_{44}+\widetilde{A}_{34}\widetilde{A}_{43}) -\widetilde{A}_{23}\widetilde{A}_{32}\widetilde{A}_{44}\}\\&\quad +\widetilde{A}_{12}\{\widetilde{A}_{21}(\widetilde{A}_{33}\widetilde{A}_{44}+\widetilde{A}_{34}\widetilde{A}_{43}) +\widetilde{A}_{23}\widetilde{A}_{31}\widetilde{A}_{44}\}\\&\quad -\widetilde{A}_{13}(\widetilde{A}_{21}\widetilde{A}_{32}\widetilde{A}_{44}+\widetilde{A}_{22}\widetilde{A}_{31}\widetilde{A}_{44}) \end{aligned}$$

with

$$\begin{aligned}&\widetilde{A}_{11}=\frac{r_p}{(1+\gamma c_p\widetilde{P}_*\widetilde{E}_*)^2}\left( 1-\frac{\widetilde{P}_*(2+\gamma c_p\widetilde{P}_*\widetilde{E}_*)}{K_p}\right) -\frac{\beta a\widetilde{Z}_*}{(a+\widetilde{P}_*)^2}, \ \widetilde{A}_{12}=\frac{\beta \widetilde{P}_*}{a+\widetilde{P}_*},\\&\widetilde{A}_{13}=\frac{r_p\gamma c_p\widetilde{P}^2_*}{(1+\gamma c_p\widetilde{P}_*\widetilde{E}_*)^2}\left( 1-\frac{\widetilde{P}_*}{K_p}\right) , \ \widetilde{A}_{21}=\frac{a\beta c_z \widetilde{Z}_*}{(a+\widetilde{P}_*)^2}, \ \widetilde{A}_{22}=\frac{r_z\widetilde{Z}_*}{K_z}, \\&\widetilde{A}_{23}=\gamma c_e\widetilde{Z}_*,\\&\widetilde{A}_{31}=\widetilde{A}_{32}=\gamma \widetilde{E}_*, \ \widetilde{A}_{33}=d_e+\gamma (\widetilde{P}_*+\widetilde{Z}_*), \ \widetilde{A}_{34}=\frac{r_ebp}{(b+\widetilde{M}_*)^2}, \\&\widetilde{A}_{43}=\frac{\phi }{(1+c\widetilde{E}_*)^2}, \widetilde{A}_{44}=\phi _0. \end{aligned}$$