Abstract
The present paper aims to study the interaction of toxin producing phytoplankton (TPP)–zooplankton (a prey–predator interaction) and its role in plankton ecology. The delay in the zooplankton predation is considered and its effect on the overall dynamic of phytoplankton–zooplankton interaction is studied. Moreover, Holling IV type response function is used for zooplankton grazing to account for the effect of toxication by the TPP population. It is shown that time delay can destabilise the given system and induce oscillation in population due to Hopf-bifurcation. Further stability of the bifurcating periodic solution is determined by using normal form theory and centre manifold arguments. Some numerical simulations are executed to validate the analytical findings.
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Acknowledgments
Authors are thankful to anonymous reviewers for their careful reading, useful comments and constructive suggestions for the improvement of the present research work. Further authors gratefully acknowledge the support provided by the Punjab Technical University, Kapurthala, Punjab, India.
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Amit Sharma is a Research Scholar in PTU, Kapurthala, India.
Appendix
Appendix
In order to compute the properties of the Hopf-bifurcation, we use the following transformation, \(u_{1}=P-P_*\), \(u_{2}=Z-Z_*\) and using the Taylor series expansion about \(E_*=(P_*,Z_*)\), the given system (2.1) can be rewritten as,
where
The coefficients of the nonlinear terms are given by
Let \(\tau =\tau _{k}+\mu \) , \(\bar{u}_{i}(t)=u_{i}(\tau t)\) and \(u_{t}(\theta )=u(t+\theta )\) for \(\theta \in [-1,0]\) and dropping the bars for simplification of notations, system (8.1) becomes a functional differential equation in \(C=C([-1,0],\mathfrak {R}^2)\) as
where \(u(t)=(u_1(t),u_2(t))^{T}\in \mathfrak {R}^{2}\) and \(L_{\mu }:C\rightarrow \mathfrak {R}^{3}, f:\mathfrak {R}\times C\rightarrow \mathfrak {R}^{2}\) are given, respectively, by
and
By Riesz representation theorem, there exists a function \(\phi (\theta ,\mu )\) of bounded variation for \(\theta \in [-1,0]\) such that
In fact, we can choose
where \(\delta \) denote the Dirac delta function.
For \(\phi \in C([-1,0],\mathfrak {R}^2)\), define
and
Then system (8.2) is equivalent to
For \(\psi \in C^{1}([0,1],(\mathfrak {R}^{2})^*)\), define
and a bilinear inner product
where \(\varsigma (\theta )=\varsigma (\theta ,0)\). Then \(A(0)\) and \(A^{*}\) are adjoint operators. From the results of last section, we know that \(\pm \iota \omega _{0}\tau _{k}\) are the eigenvalues of \(A(0)\) and \(\mp \iota \omega _{0}\tau _{k}\) are the eigenvalues of \(A^*\).
Let \(q(\theta )=(1,\alpha )^Te^{\iota \omega _{0}\tau _{k}\theta }\) be the eigenvector of \(A(0)\) corresponding to the eigenvalue \(\iota \omega _{0}\tau _k\) and \(q^{*}(s)=D(1,\beta )e^{\iota \omega _{0}\tau _{k}s}\) be the eigenvector of \(A^{*}\) corresponding to the eigenvalue \(-\iota \omega _{0}\tau _k\), then
It follows from the definition of \(A(0)\), \(L_{\mu }(\phi )\) and \(\eta (\theta , \mu )\) that
where I is identity matrix of order 2, that is,
Solving the above equations, we get
From the definition of \(A^*\), we obtain
Since \(q(\theta )\) is the eigenvector of \(A^*\) corresponding to -\(\iota \omega _0\tau _k\), then we have
where I is identity matrix of order 2, that is,
Solving the above equations, we get
Now,
In order to have \((q^*, q)=1\), we can choose \(D={\displaystyle \frac{1}{1+\bar{\alpha }\beta +\bar{\alpha }a_{01}\tau _{k}e^{\iota \omega _{0}\tau _{k}}}}\). such that \((\psi , A\phi )=(A^*\psi , \phi )\), or \(-\iota \omega _0(q^*, \bar{q})=(q^*, A\bar{q})=(A^*q^*, \bar{q})=(-\iota \omega _0q^*, \bar{q})=\iota \omega _0(q^*, \bar{q})\) i.e. \((q^*, \bar{q})=0\).
Next we will compute the coordinates to describe the centre manifold \(C_{0}\) at \(\mu =0\). Let \(u_{t}\) be the solution of (8.7) when \(\mu =0\).
Define
On the centre manifold \(C_{0}\), we have
where
z and \(\bar{z}\) are local coordinates for centre manifold \(C_{0}\) in the direction of \(q\) and \(q^*\) respectively. Here we consider only real solutions as W is real if \(u_{t}\) is real. For the solution \(u_{t}\in C_{0}\) of (8.7), since \(\mu =0\), we have
This equation can be rewritten as
where
From (8.9) and (8.10), we have
Now from (8.4) and (8.11), it follows that
or
where
Comparing coefficients of (8.11) and (8.14), we find
Since \(W_{20}\) and \(W_{11}\) are in \(g_{21}\), we still to compute them.
Now from (8.7) and (8.9), we have
where
Substituting (8.17) into (8.16) and comparing the coefficients, we get
From (8.16) and for \(\theta \in [-1,0)\)
Using (8.11) in (8.19) and comparing coefficients with (8.17), we can obtain
and
From the definition of \(A(0)\), (8.18) and (8.20), we obtain
Solving it and for \(q(\theta )=(1,q_1,q_2)^Te^{\iota \omega _0\tau _k\theta }\), we have
Similarly, from (8.18) and (8.21) it follows that,
where \(E_{1}=({E_{1}}^{(1)},{E_{1}}^{(2)})^{T}\) and \(E_{2}=({E_{2}}^{(1)},{E_{2}}^{(2)})^{T}\) are three dimensional constant vectors, and can be determined by setting \(\theta =0\) in \(H(z,\bar{z},\theta )\).
Again from the definition of \(A(0)\) and (8.18), we have
and
where \(\varsigma (\theta )=\varsigma (0,\theta )\).
From (8.16), we know when \(\theta =0\),
That is,
From (8.9), we have
Thus we can obtain,
Comparing the coefficients in (8.26) and using (8.27), we get
Since \(\iota \omega _{0}\tau _{k}\) is the eigenvalue of \(A(0)\) corresponding to eigenvector \(q(0)\), then
Substituting (8.22) and (8.28) into (8.24), we find
or
or \(E_1=2\left[ \begin{array}{cc} &{}p_1 \\ &{}q_1 \end{array}\right] \left[ \begin{array}{cccc} &{}2\iota \omega _{0}-a_{10} &{} -a_{01}e^{-2\iota \omega \tau _0}\\ &{}-b_{10} &{} 2\iota \omega _{0}-b_{01} \end{array}\right] ^{-1}\)
Similarly substituting (8.23) and (8.29) into (8.25), we obtain
or
Thus, we can determine \(W_{20}(\theta )\), \(W_{11}(\theta )\) from (8.22), (8.23) and \(g_{21}\) can be computed from (8.15).
Finally, we can compute the following quantities:
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Sharma, A., Sharma, A.K. & Agnihotri, K. Analysis of a toxin producing phytoplankton–zooplankton interaction with Holling IV type scheme and time delay. Nonlinear Dyn 81, 13–25 (2015). https://doi.org/10.1007/s11071-015-1969-5
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DOI: https://doi.org/10.1007/s11071-015-1969-5
Keywords
- Phytoplankton–zooplankton
- Holling IV type scheme
- Time delay
- Hopf-bifurcation
- Normal form theory
- Centre manifold theorem