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

  • Ashok MondalEmail author
  • A. K. Pal
  • G. P. Samanta
Correction
  • 39 Downloads

1 Correction to: International Journal of Dynamics and Control  https://doi.org/10.1007/s40435-018-0501-4

In the original publication, Theorem 4.6 has been published incorrectly. The corrected theorem is given below:

Theorem 4.6

Suppose that system (3.2) satisfies the following conditions:
$$\begin{aligned}&(i) \ k_2-\frac{\beta _{21}k_1}{\alpha _1}>0 \ \ \text {and / or}\ \frac{\gamma _1k_1}{\alpha _1}-\delta>0; \\&(ii)\ k_1-\frac{\beta _{12}k_2}{\alpha _2}>0;\\&(iii)\ \gamma _1\hat{X_1}-\delta -\gamma _2\hat{X_2}>0;\\&(iv)\ k_2-\beta _{21}\tilde{X_1}-\gamma _2\tilde{Y}>0 , \end{aligned}$$
then system (3.2) is permanence.

Proof

Let us consider the average Lyapunov function in the form \(V(X_1,X_2,Y)=X_1^{\theta _{1}}X_2^{\theta _{2}}Y^{\theta _{3}}\) where each \(\theta _{i}\ (i=1,2,3)\) is assumed to be positive. In the interior of \(\mathbb {R}_{+}^{3}\), we have
$$\begin{aligned} \frac{\dot{V}}{V}= & {} \psi (X_1,X_2,Y)=\theta _{1}\left[ k_1-\alpha _1X_1 -\beta _{12}X_2-\gamma _1Y\right] \\&+\,\theta _{2}\left[ k_2-\alpha _2X_2-\beta _{21}X_1-\gamma _2Y\right] \\&+\,\theta _{3}\left[ \gamma _1X_1-\delta -\gamma _2X_2\right] . \end{aligned}$$
To prove the permanence of the system, we shall have to show that \(\psi (X_1,X_2,Y)>0\), for all boundary equilibria of the system. The values of \(\psi (X_1,X_2,Y)\), at the boundary equilibria \(E_{0}, E_{1}, E_{2}, E_{3} \ \text{ and } \ E_{4}\), are the following:
$$\begin{aligned} \begin{array}{lcl} E_{0}&{}:&{}\theta _{1}k_1+\theta _{2}k_2-\theta _3\delta .\\ E_{1}&{}:&{}\theta _{2}\left( k_2-\frac{\beta _{21}k_1}{\alpha _1}\right) +\theta _3\left( \frac{\gamma _1k_1}{\alpha _1}-\delta \right) .\\ E_{2}&{}:&{}\theta _{1}\left( k_1-\frac{\beta _{12}k_2}{\alpha _2}\right) +\theta _3\left( -\delta -\frac{\gamma _2k_2}{\alpha _2}\right) .\\ E_{3}&{}:&{}\theta _{3}\left\{ \gamma _1\hat{X_1}-\delta -\gamma _2\hat{X_2}\right\} .\\ E_{4}&{}:&{}\theta _{2}\left\{ k_2-\beta _{21}\tilde{X_1} -\gamma _2\tilde{Y} \right\} . \end{array} \end{aligned}$$
Now, \(\psi (0,0,0)>0\) is automatically satisfied for some \(\theta _{i}>0\ (i=1,2,3)\). Also, if the inequalities \((i)-(iv)\) hold, \(\psi \) is positive at \(E_{1}, E_{2}, E_{3} \ \text{ and } \ E_{4}\) for some \(\theta _i > 0\)\((i = 1, 2, 3).\) Therefore, system (3.2) is permanence [1] if the conditions (i) – (iv) are fulfilled. Hence the theorem. \(\square \)

Reference

  1. 1.
    Freedman HI, Ruan S (1995) Uniform persistence in functional differential equations. J Differ Equ 115:173–192MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Engineering Science and TechnologyShibpur, HowrahIndia
  2. 2.Department of MathematicsS. A. Jaipuria CollegeKolkataIndia

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