Rich dynamics of non-toxic phytoplankton, toxic phytoplankton and zooplankton system with multiple gestation delays

Abstract

The plankton community is classified into three category of species, namely, non-toxic phytoplankton (NTP), toxic phytoplankton (TPP) and zooplankton. In this work we have introduced a mathematical model for the interaction of NTP, TPP and zooplankton population in an open marine system. We have incorporated two time delays and observed important mathematical characteristics of the proposed model such as positivity, boundedness, stability and Hopf-bifurcation for all possible combinations of both the delays at the interior equilibrium point of the model system. It is noted that increase in gestation delay may lead to the destabilization of stationary points through the creation of limit cycles. Various numerical simulations are performed to validate the analytical findings obtained here.

Appendices

Appendix A

Proof of the Proposition

Proof

Now we study the local stability behaviour of the equilibrium points by computing corresponding variational matrix:

$$\begin{aligned}&\displaystyle V(X_1,X_2,Y)\nonumber \\&\quad = \left[ \begin{array}{ccc} k_1-2\alpha _1X_1-\beta _{12}X_2-\gamma _1Y&{}\quad -\beta _{12}X_1&{}\quad -\gamma _1X_1\\ - \beta _{21}X_2 &{}\quad k_2-2\alpha _2X_2-\beta _{21}X_1-\gamma _2Y&{}\quad -\gamma _2X_2 \\ \gamma _1Y&{}\quad -\gamma _2Y &{}\quad \gamma _1X_1-\delta -\gamma _2X_2\\ \end{array} \right] \end{aligned}$$

(A.1)

After substituting $E_i = (X_1,X_2,Y); i = 0, 1, 2, 3$ into (A.1), we obtain the eigenvalues for each equilibrium point: (A) $E_0 = (0, 0, 0)$ is always unstable since eigenvalues associated with (A.1) at $E_0$ are as follows:

$$\begin{aligned} \displaystyle \lambda _1=k_1(>0), \ \lambda _2=k_2(>0) \ \text {and}\ \lambda _3=-\delta (<0). \end{aligned}$$

(B) $E_1 = (\frac{k_1}{\alpha _1}, 0, 0)$ is locally asymptotically stable if $k_2<\frac{\beta _{21}k_1}{\alpha _1}$ and $\frac{\gamma _1k_1}{\alpha _1}<\delta$, thereafter eigenvalues associated with (A.1) at $E_1$ are given by

$$\begin{aligned}&\displaystyle \lambda _1=-k_1(<0), \ \lambda _2=k_2-\frac{\beta _{21}k_1}{\alpha _1}(<0) \\&\quad \text {and}\ \lambda _3=\frac{\gamma _1k_1}{\alpha _1}-\delta (<0). \end{aligned}$$

(C) $E_2 = \left( 0,\frac{k_2}{\alpha _2},0\right)$ is locally asymptotically stable if $k_1<\frac{\beta _{12}k_2}{\alpha _2}$. For this situation the eigenvalues associated with (A.1) at $E_2$ are given by

$$\begin{aligned}&\displaystyle \lambda _1=-k_2(<0), \ \lambda _2=k_1-\frac{\beta _{12}k_2}{\alpha _2}(<0),\\&\quad \text {and}\ \lambda _3=-\frac{\gamma _2k_2}{\alpha _2}-\delta (<0). \end{aligned}$$

(D) $E_3 = ({\hat{X}}_1,{\hat{X}}_2,0)$ is locally asymptotically stable if $\gamma _1{\hat{X}}_1<\delta +\gamma _2{\hat{X}}_2\ \hbox {and}\ A_{1}>0, B_{1}>0$. The eigenvalues associated with (A.1) at $E_3$ are given by

$$\begin{aligned} \displaystyle \lambda _3=\gamma _1{\hat{X}}_1-\delta -\gamma _2{\hat{X}}_2,(<0, \ \text {if}\ \gamma _1{\hat{X}}_1<\delta +\gamma _2{\hat{X}}_2) \end{aligned}$$

and the other two eigenvalues are the roots of the equation:

$$\begin{aligned} \displaystyle \lambda ^2+A_1\lambda +B_1=0 \end{aligned}$$

where

$$\begin{aligned} \displaystyle A_1= & {} -k_1+2\alpha _1{\hat{X}}_1+\beta _{12}{\hat{X}}_2-k_2+2\alpha _2{\hat{X}}_2+\beta _{21}{\hat{X}}_1 \ \text {and}\\ B_1= & {} (k_1-2\alpha _1{\hat{X}}_1-\beta _{12}{\hat{X}}_2) (k_2-2\alpha _2{\hat{X}}_2\\&-\beta _{21}{\hat{X}}_1)-\beta _{12}\beta _{21}{\hat{X}}_1{\hat{X}}_2. \end{aligned}$$

Then all the roots of $\lambda ^2+A_1\lambda +B_1=0$ are negative or have negative real parts if $2(\alpha _1{\hat{X}}_1+\alpha _2{\hat{X}}_2)+\beta _{12}{\hat{X}}_2+\beta _{21}{\hat{X}}_1>k_1+k_2$ and $(k_1-2\alpha _1{\hat{X}}_1-\beta _{12}{\hat{X}}_2) (k_2-2\alpha _2{\hat{X}}_2-\beta _{21}{\hat{X}}_1)>\beta _{12}\beta _{21}{\hat{X}}_1{\hat{X}}_2$. Therefore, $E_3$ exists and is locally asymptotically stable.

(E) $E_4 = (\tilde{X_1},0,{\tilde{Y}})$ is locally asymptotically stable if $\beta _{21}\tilde{X_1}+\gamma _2{\tilde{Y}}>k_2\ \hbox {and}\ A_{2}>0, B_{2}>0$. The eigenvalues associated with (A.1) at $E_3$ are given by

$$\begin{aligned} \displaystyle \lambda _3= k_2-\beta _{21}\tilde{X_1}-\gamma _2{\tilde{Y}},(<0, \ \text {if}\ \beta _{21}\tilde{X_1}+\gamma _2{\tilde{Y}}>k_2) \end{aligned}$$

and the other two eigenvalues are the roots of the equation:

$$\begin{aligned} \displaystyle \lambda ^2+A_2\lambda +B_2=0 \end{aligned}$$

where

$$\begin{aligned} \displaystyle A_2= & {} 2\alpha _1\tilde{X_1}+\gamma _1{\tilde{Y}}+\delta -k_1-\gamma _1\tilde{X_1} \ \text {and}\\ B_2= & {} k_1\gamma _1\tilde{X_1}+2\alpha _1\tilde{X_1}\delta +\gamma _1\delta {\tilde{Y}}-k_1\delta -2\alpha _1\gamma _1\tilde{X_1}^2. \end{aligned}$$

Then all the roots of $\lambda ^2+A_2\lambda +B_2=0$ are negative or have negative real parts if $2\alpha _1\tilde{X_1}+\gamma _1{\tilde{Y}}+\delta >k_1+\gamma _1\tilde{X_1}$ and $k_1\gamma _1\tilde{X_1}+2\alpha _1\tilde{X_1}\delta +\gamma _1\delta {\tilde{Y}}>k_1\delta +2\alpha _1\gamma _1\tilde{X_1}^2$. Therefore, $E_4$ exists and is locally asymptotically stable.

(F) Again, the Jacobian matrix of the model (3.2) at its interior equilibrium point $E^*(X_1^*,X_2^*,Y)$ can be written as follows:

$$\begin{aligned} \displaystyle V(E^*)= \left[ \begin{array}{ccc} -\alpha _1X_1^*&{} \quad -\beta _{12}X_1^*&{}\quad -\gamma _1X_1^*\\ - \beta _{21}X_2^* &{}\quad -\alpha _2X_2^*&{}\quad -\gamma _2X_2^* \\ \gamma _1Y^*&{}\quad -\gamma _2Y^* &{}\quad 0\\ \end{array} \right] \end{aligned}$$

The corresponding characteristic equation is given by

$$\begin{aligned} \displaystyle \lambda ^3+A_{2}\lambda ^2+A_{3}\lambda +A_{4}=0, \end{aligned}$$

where

$$\begin{aligned} A_2= & {} \alpha _1X_1^*+\alpha _2X_2^*(>0),\\ A_3= & {} (\alpha _1\alpha _2-\beta _{12}\beta _{21})X_1^*X_2^*-\gamma _2^2X_2^*Y^*+\gamma _1^2X_1^*Y^*,\\ A_4= & {} \left( -\alpha _1\gamma _2^2+\alpha _2\gamma _1^2-\beta _{12}\gamma _1\gamma _2+\gamma _2\gamma _1\beta _{21}\right) X_1^*X_2^*Y^*. \end{aligned}$$

By Routh Hurwitz’s criterion, all the eigenvalues of $V(E^*)$ have negative real parts if

$$\begin{aligned}&(i)\quad \alpha _2\gamma _1^2+\gamma _2\gamma _1\beta _{21}>\alpha _1\gamma _2^2+\beta _{12}\gamma _1\gamma _2\\&(ii)\quad A_{2}A_{3}-A_{4}>0. \end{aligned}$$

Therefore, $E^*$ exists and is locally asymptotically stable if $\alpha _2\gamma _1^2+\gamma _2\gamma _1\beta _{21}>\alpha _1\gamma _2^2+\beta _{12}\gamma _1\gamma _2\ \hbox {and}\ A_{2}A_{3}-A_{4}>0$. $\square$