# American Institute of Mathematical Sciences

• Previous Article
Existence and well-posedness for excess demand equilibrium problems
• NACO Home
• This Issue
• Next Article
A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings
doi: 10.3934/naco.2021036
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Dynamical complexity in a delayed Plankton-Fish model with alternative food for predators

 1 Research scholar I.K.Gujral Punjab Technical University, Jalandhar, Punjab, India 2 P.G.Department of Mathematics, Khalsa College Amritsar, Punjab, India 3 Department of Applied Sciences, D.A.V.Institute of Engineering and Technology, Jalandhar, Punjab, India 4 Department of Mathematics, L.R.D.A.V.College, Jagraon, Punjab, India

* Corresponding author: Rajinder Pal Kaur

This paper is handled by Kapil Kumar Sharma as the guest editor

Received  December 2020 Revised  August 2021 Early access September 2021

The present manuscript deals with a 3-D food chain ecological model incorporating three species phytoplankton, zooplankton, and fish. To make the model more realistic, we include predation delay in the fish population due to the vertical migration of zooplankton species. We have assumed that additional food is available for both the predator population, viz., zooplankton, and fish. The main motive of the present study is to analyze the impact of available additional food and predation delay on the plankton-fish dynamics. The positivity and boundedness (with and without delay) are proved to make the system biologically valid. The steady states are determined to discuss the stability behavior of non-delayed dynamics under certain conditions. Considering available additional food as a control parameter, we have estimated ranges of alternative food for maintaining the sustainability and stability of the plankton-fish ecosystem. The Hopf-bifurcation analysis is carried out by considering time delay as a bifurcation parameter. The predation delay includes complexity in the system dynamics as it passes through its critical value. The direction of Hopf-bifurcation and stability of bifurcating periodic orbits are also determined using the centre manifold theorem. Numerical simulation is executed to validate theoretical results.

Citation: Rajinder Pal Kaur, Amit Sharma, Anuj Kumar Sharma. Dynamical complexity in a delayed Plankton-Fish model with alternative food for predators. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021036
##### References:

show all references

##### References:
Existence of steady states $V_1$, $V_2$ and $V_3$
stability of the system around $V^*$ at $r_5 = 0.7$
Existence of Hopf-bifurcation at $r_5 = 0.8$
Bifurcation diagram for $0.1<r_5\leq 1$
Existence of stability at $\tau = 0$ (left fig.) and $\tau = 1.4<\tau_0 = 1.5$ (right fig.)
Existence of Hopf-bifurcation at $\tau = 1.5 = \tau_0$, occurrence of limit cycles at $\tau = 1.6, 2, 3$(right fig)
Bifurcation diagrams for $1\leq\tau\leq3$
with $r_3$ on x-axis and $r_5$, P(t), Z(t) and F(t) on y-axis">Figure 8.  Co-existence of all species according to table 2 with $r_3$ on x-axis and $r_5$, P(t), Z(t) and F(t) on y-axis
Biological interpretation of Parameters
 Parameter Biological interpretation $r_1$ Growth rate of Phytoplankton. $\alpha_1$ Death rate phytoplankton. $\beta_1$ Maximum capture rate of phytoplankton by zooplankton. $\beta_2$ Maximum conversion rate of zooplankton. $\gamma_1$ Half saturation constant. $\gamma_2$ Half saturation constant. $r_5$ Additional food available for zooplankton. $r_2$ Death rate of zooplankton. $a_1$ Maximum capture rate of zooplankton by fish. $a_2$ Maximum conversion rate of fish. $\theta$ Rate of toxin produced by TPP. $r_3$ Additional food available for fish. $\alpha_2$ Rate of quadratic harvesting. $r_4$ Natural mortality rate of fish.
 Parameter Biological interpretation $r_1$ Growth rate of Phytoplankton. $\alpha_1$ Death rate phytoplankton. $\beta_1$ Maximum capture rate of phytoplankton by zooplankton. $\beta_2$ Maximum conversion rate of zooplankton. $\gamma_1$ Half saturation constant. $\gamma_2$ Half saturation constant. $r_5$ Additional food available for zooplankton. $r_2$ Death rate of zooplankton. $a_1$ Maximum capture rate of zooplankton by fish. $a_2$ Maximum conversion rate of fish. $\theta$ Rate of toxin produced by TPP. $r_3$ Additional food available for fish. $\alpha_2$ Rate of quadratic harvesting. $r_4$ Natural mortality rate of fish.
Impact of additional food ($r_3$ and $r_5$) on the co-existence of species
 $r_3$ $r_5$ P(t) Z(t) F(t) 0.01 0.01 23.8297 3.5257 6.6514 0.05 0.01 25.1787 2.4806 6.7049 0.08 0.01 26.0261 1.7771 6.8449 $0.1$ 0.01 26.6036 1.2770 6.9687 $0.2$ $0.01$ 27.9978 0.0000 11.1111 $0.2$ $0.4$ 27.8282 0.1586 11.9851 $0.2$ $0.9$ 26.0543 1.7483 20.0424 $0.2$ $1$ 25.6022 2.1321 21.8148 $0.2$ $1.1$ 25.0996 2.5444 23.6513
 $r_3$ $r_5$ P(t) Z(t) F(t) 0.01 0.01 23.8297 3.5257 6.6514 0.05 0.01 25.1787 2.4806 6.7049 0.08 0.01 26.0261 1.7771 6.8449 $0.1$ 0.01 26.6036 1.2770 6.9687 $0.2$ $0.01$ 27.9978 0.0000 11.1111 $0.2$ $0.4$ 27.8282 0.1586 11.9851 $0.2$ $0.9$ 26.0543 1.7483 20.0424 $0.2$ $1$ 25.6022 2.1321 21.8148 $0.2$ $1.1$ 25.0996 2.5444 23.6513
 [1] R. P. Gupta, Shristi Tiwari, Shivam Saxena. The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021160 [2] Xin-You Meng, Yu-Qian Wu, Jie Li. Bifurcation analysis of a Singular Nutrient-plankton-fish model with taxation, protected zone and multiple delays. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 391-423. doi: 10.3934/naco.2020010 [3] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [4] S. R.-J. Jang, J. Baglama, P. Seshaiyer. Intratrophic predation in a simple food chain with fluctuating nutrient. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 335-352. doi: 10.3934/dcdsb.2005.5.335 [5] Maria Paola Cassinari, Maria Groppi, Claudio Tebaldi. Effects of predation efficiencies on the dynamics of a tritrophic food chain. Mathematical Biosciences & Engineering, 2007, 4 (3) : 431-456. doi: 10.3934/mbe.2007.4.431 [6] Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275 [7] J. David Logan, William Wolesensky, Anthony Joern. Insect development under predation risk, variable temperature, and variable food quality. Mathematical Biosciences & Engineering, 2007, 4 (1) : 47-65. doi: 10.3934/mbe.2007.4.47 [8] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [9] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [10] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [11] Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503 [12] Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457 [13] Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 [14] Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 [15] Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 [16] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [17] Soliman A. A. Hamdallah, Sanyi Tang. Stability and bifurcation analysis of Filippov food chain system with food chain control strategy. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1631-1647. doi: 10.3934/dcdsb.2019244 [18] Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71 [19] R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 [20] Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197

Impact Factor:

## Metrics

• HTML views (182)
• Cited by (0)

• on AIMS