# American Institute of Mathematical Sciences

doi: 10.3934/naco.2021036
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## 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 and Optimization, doi: 10.3934/naco.2021036
##### References:
 [1] K. M. Bailey and J. T. Duffy-Anderson, Fish Predation and Mortality, Encyclopedia of Ocean Sciences, (2001), 961–968. doi: 10.1006/rwos.2001.0024. [2] S. Chakraborty and J. Chattopadhyay, Nutrient-phytoplankton-zooplankton dynamics in the presence of additional food source-a mathematical study, Journal of Biological Systems, 16 (2008), 547-564. [3] J. Chattopadhyay, R. R.Sarkar and A. Abdllaoui, A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA. J. Math. Appl. Med. Biol., 19 (2002), 137-161. [4] J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer-Verlag, Heidelberg, 1977. [5] K. Das and S. Ray, Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system, Ecol. Model, 215 (2008), 69-76. [6] J. Dhar, A. K. sharma and S. Tegar, The role of delay in digestion of plankton by fish population: A fishary model, The Journal of Nonlinear Sciences and its Applications, 1 (2008), 13-19.  doi: 10.22436/jnsa.001.01.03. [7] B. Dubey and A. Kumar, Dynamics of prey-predator model with stage structure in prey including maturation and gestation delays, Nonlinear Dynamics, 96 (2019), 2653-2679. [8] J. Ghosh, B. Sahoo and S. Poria, Prey-predator dynamics with prey refuge providing additional food to predator, Chaos, Solitons and Fractals, 96 (2017), 110-119.  doi: 10.1016/j.chaos.2017.01.010. [9] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, 1992. doi: 10.1007/978-94-015-7920-9. [10] J. K. Hale, Ordinary Differential Equations, Wiley, New Yark, 1969. [11] J. K. Hale, Theory of Functional Differential Equations, Springer, Heidelberg, 1977. [12] M. Haque and D. Greenhalgh, When a predator avoids infected prey: a model-based theoretical study, Math. Med. Biol., 27 (2010), 75-94.  doi: 10.1093/imammb/dqp007. [13] J. D. Harwood and J. J.Obrycki, The role of alternative prey in sustaining predator population, in Proceedings of Second International Symposium on Biological Control of Arthropods (ed. M.S. Hoddle), 2 (2005), 453–462. [14] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf-Bifurcation, Cambridge University Press, Cambridge, 1981. [15] G. R. Huxel and K. McCann, Food web stability: the influence of trophic flows across habitats, Am. Nat., 152 (1998), 460-469. [16] G. R. Huxel, K. McCann and G. A. Polis, Effects of partitioning allochthonous and autochthonous resources on food web stability, Ecol. Res., 17 (2002), 419-432. [17] R. P. Kaur, A. Sharma and A. K. Sharma, Complex dynamics of phytoplankton-zooplankton intercation system with predation and toxin liberation delay, International Journal of Grid And Distributing Computing, 12 (2019), 23-50. [18] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [19] B. Mukhopadhyay and R. Bhattacharyya, Role of gestation delay in a planktonfish model under stochastic fluctuations, Mathematical Biosciences, 215 (2008), 26-34.  doi: 10.1016/j.mbs.2008.05.007. [20] B. S. R. V. Prasad, M. Banerjee and P. D. N. Srinivasu, Dynamics of additional food provided predator-prey system with mutually interfering predators, Math. Biosci., 246 (2013), 176-190.  doi: 10.1016/j.mbs.2013.08.013. [21] M. Rehim and M. Imran, Dynamical analysis of a delay model of phytoplankton-zooplankton interaction, Applied Mathematical Modelling, 36 (2002), 638-647.  doi: 10.1016/j.apm.2011.07.018. [22] S. Ruan, The effect of delays on stability and persistence in plankton models, Nonlinear Analalysis, 24 (1995), 575-585.  doi: 10.1016/0362-546X(95)93092-I. [23] M. W. Sabelis and P. C. J. V. Rijn, When does alternative food promote biological pest control?, In Proc. Second Int. Symp. Biol. Control of Arthropods (ed. Hoddle MS), 2 (2005), 428–437. [24] T. Saha and M. Bandyopadhyay, Dynamical analysis of toxin producing Phytoplankton-Zooplankton interactions, Nonlinear Analysis: Real World Applications, 10 (2009), 314-332.  doi: 10.1016/j.nonrwa.2007.09.001. [25] B. Sahoo, Effects of Additional Foods to Predators on Nutrient-Consumer-Predator Food Chain Model, Isrnbio Mathematics, 2012. doi: 10.5402/2012/796783. [26] B. Sahoo and S. Poria, Disease control in a food chain model supplying alternative food, Appl. Math. Model, 37 (2013), 5653-5663.  doi: 10.1016/j.apm.2012.11.017. [27] M. Sen, P. D. N. Srinivasu and M. Banerjeea, Global dynamics of an additional food provided predator-prey system with constant harvest in predators, Applied Mathematics and Computaions, 250 (2015), 193-211.  doi: 10.1016/j.amc.2014.10.085. [28] A. Sharma, A. K. Sharma and K. Agnihotry, The dynamic of plankton-nutrien interaction with delay, Applied Mathematics and Computation, 231 (2014), 503-515.  doi: 10.1016/j.amc.2014.01.042. [29] A. Sharma, A. K. Sharma and K. Agnihotri, Analysis of a toxin producing phytoplankton-zooplankton interaction with Holling IV type scheme and time delay, Nonlinear Dynamics, 81 (2015), 13-25.  doi: 10.1007/s11071-015-1969-5. [30] A. K. Sharma, A. Sharma and K. Agnihotry, Complex dynamic of plankton-fish interaction with quadratic harvesting and time delay, Model Earth Syst. Environ., 2 (2016), 1-17. [31] A. K. Sharma, A. Sharma and K. Agnihotry, Bifurcation behaviors analysis of a plankton model with multiple delays, International Journal of Biomathematics, 9 (2016), 1650086 (25 pages). doi: 10.1142/S1793524516500868. [32] P. D. N Srinivasu, B. S. R. V. Prasad and M. Venkatesulu, Biological control through provision of additional food to predators: a theoretical study, Theor. Popul. Biol., 72 (2007), 111-120. [33] P. D. N. Srinivasu and B. S. R. V. Prasad, Time optimal control of an additional food provided predator-prey system with applications to pest management and biological conservation, J. Math. Biol., 60 (2010), 591-613.  doi: 10.1007/s00285-009-0279-2. [34] P. D. N. Srinivasu and B. S. R. V. Prasad, Role of quantity of additional food to predators as a control in predator-prey systems with relevance to pestmanagement and biological conservation, Bull. Math. Biol., 73 (2011), 2249-2276.  doi: 10.1007/s11538-010-9601-9. [35] D. Stiefs, G. A. K. van Voorn, B. W. Kooi, U. Feudel and T. Gross, Food quality in producer-grazer models: A generalized analysis, Am. Nat., 176 (2010), 367-380.  doi: 10.1086/655429. [36] J. N. Van Baalen, V. Krivan, P. C. J. Van Rijn and M. W. Sabelis, Alternative food, switching predators, and the persistence of predator-prey systems, Am. Nat., 157 (2001), 512-524.  doi: 10.1086/319933. [37] P. C. J. Van Rijn, Y. M. Van Houten and M. W. Sabelis, How plants benefit from providing food to predators even when it is also edible to herbivores, Ecology, 83 (2002), 2664-2679.

show all references

##### References:
 [1] K. M. Bailey and J. T. Duffy-Anderson, Fish Predation and Mortality, Encyclopedia of Ocean Sciences, (2001), 961–968. doi: 10.1006/rwos.2001.0024. [2] S. Chakraborty and J. Chattopadhyay, Nutrient-phytoplankton-zooplankton dynamics in the presence of additional food source-a mathematical study, Journal of Biological Systems, 16 (2008), 547-564. [3] J. Chattopadhyay, R. R.Sarkar and A. Abdllaoui, A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA. J. Math. Appl. Med. Biol., 19 (2002), 137-161. [4] J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer-Verlag, Heidelberg, 1977. [5] K. Das and S. Ray, Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system, Ecol. Model, 215 (2008), 69-76. [6] J. Dhar, A. K. sharma and S. Tegar, The role of delay in digestion of plankton by fish population: A fishary model, The Journal of Nonlinear Sciences and its Applications, 1 (2008), 13-19.  doi: 10.22436/jnsa.001.01.03. [7] B. Dubey and A. Kumar, Dynamics of prey-predator model with stage structure in prey including maturation and gestation delays, Nonlinear Dynamics, 96 (2019), 2653-2679. [8] J. Ghosh, B. Sahoo and S. Poria, Prey-predator dynamics with prey refuge providing additional food to predator, Chaos, Solitons and Fractals, 96 (2017), 110-119.  doi: 10.1016/j.chaos.2017.01.010. [9] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, 1992. doi: 10.1007/978-94-015-7920-9. [10] J. K. Hale, Ordinary Differential Equations, Wiley, New Yark, 1969. [11] J. K. Hale, Theory of Functional Differential Equations, Springer, Heidelberg, 1977. [12] M. Haque and D. Greenhalgh, When a predator avoids infected prey: a model-based theoretical study, Math. Med. Biol., 27 (2010), 75-94.  doi: 10.1093/imammb/dqp007. [13] J. D. Harwood and J. J.Obrycki, The role of alternative prey in sustaining predator population, in Proceedings of Second International Symposium on Biological Control of Arthropods (ed. M.S. Hoddle), 2 (2005), 453–462. [14] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf-Bifurcation, Cambridge University Press, Cambridge, 1981. [15] G. R. Huxel and K. McCann, Food web stability: the influence of trophic flows across habitats, Am. Nat., 152 (1998), 460-469. [16] G. R. Huxel, K. McCann and G. A. Polis, Effects of partitioning allochthonous and autochthonous resources on food web stability, Ecol. Res., 17 (2002), 419-432. [17] R. P. Kaur, A. Sharma and A. K. Sharma, Complex dynamics of phytoplankton-zooplankton intercation system with predation and toxin liberation delay, International Journal of Grid And Distributing Computing, 12 (2019), 23-50. [18] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [19] B. Mukhopadhyay and R. Bhattacharyya, Role of gestation delay in a planktonfish model under stochastic fluctuations, Mathematical Biosciences, 215 (2008), 26-34.  doi: 10.1016/j.mbs.2008.05.007. [20] B. S. R. V. Prasad, M. Banerjee and P. D. N. Srinivasu, Dynamics of additional food provided predator-prey system with mutually interfering predators, Math. Biosci., 246 (2013), 176-190.  doi: 10.1016/j.mbs.2013.08.013. [21] M. Rehim and M. Imran, Dynamical analysis of a delay model of phytoplankton-zooplankton interaction, Applied Mathematical Modelling, 36 (2002), 638-647.  doi: 10.1016/j.apm.2011.07.018. [22] S. Ruan, The effect of delays on stability and persistence in plankton models, Nonlinear Analalysis, 24 (1995), 575-585.  doi: 10.1016/0362-546X(95)93092-I. [23] M. W. Sabelis and P. C. J. V. Rijn, When does alternative food promote biological pest control?, In Proc. Second Int. Symp. Biol. Control of Arthropods (ed. Hoddle MS), 2 (2005), 428–437. [24] T. Saha and M. Bandyopadhyay, Dynamical analysis of toxin producing Phytoplankton-Zooplankton interactions, Nonlinear Analysis: Real World Applications, 10 (2009), 314-332.  doi: 10.1016/j.nonrwa.2007.09.001. [25] B. Sahoo, Effects of Additional Foods to Predators on Nutrient-Consumer-Predator Food Chain Model, Isrnbio Mathematics, 2012. doi: 10.5402/2012/796783. [26] B. Sahoo and S. Poria, Disease control in a food chain model supplying alternative food, Appl. Math. Model, 37 (2013), 5653-5663.  doi: 10.1016/j.apm.2012.11.017. [27] M. Sen, P. D. N. Srinivasu and M. Banerjeea, Global dynamics of an additional food provided predator-prey system with constant harvest in predators, Applied Mathematics and Computaions, 250 (2015), 193-211.  doi: 10.1016/j.amc.2014.10.085. [28] A. Sharma, A. K. Sharma and K. Agnihotry, The dynamic of plankton-nutrien interaction with delay, Applied Mathematics and Computation, 231 (2014), 503-515.  doi: 10.1016/j.amc.2014.01.042. [29] A. Sharma, A. K. Sharma and K. Agnihotri, Analysis of a toxin producing phytoplankton-zooplankton interaction with Holling IV type scheme and time delay, Nonlinear Dynamics, 81 (2015), 13-25.  doi: 10.1007/s11071-015-1969-5. [30] A. K. Sharma, A. Sharma and K. Agnihotry, Complex dynamic of plankton-fish interaction with quadratic harvesting and time delay, Model Earth Syst. Environ., 2 (2016), 1-17. [31] A. K. Sharma, A. Sharma and K. Agnihotry, Bifurcation behaviors analysis of a plankton model with multiple delays, International Journal of Biomathematics, 9 (2016), 1650086 (25 pages). doi: 10.1142/S1793524516500868. [32] P. D. N Srinivasu, B. S. R. V. Prasad and M. Venkatesulu, Biological control through provision of additional food to predators: a theoretical study, Theor. Popul. Biol., 72 (2007), 111-120. [33] P. D. N. Srinivasu and B. S. R. V. Prasad, Time optimal control of an additional food provided predator-prey system with applications to pest management and biological conservation, J. Math. Biol., 60 (2010), 591-613.  doi: 10.1007/s00285-009-0279-2. [34] P. D. N. Srinivasu and B. S. R. V. Prasad, Role of quantity of additional food to predators as a control in predator-prey systems with relevance to pestmanagement and biological conservation, Bull. Math. Biol., 73 (2011), 2249-2276.  doi: 10.1007/s11538-010-9601-9. [35] D. Stiefs, G. A. K. van Voorn, B. W. Kooi, U. Feudel and T. Gross, Food quality in producer-grazer models: A generalized analysis, Am. Nat., 176 (2010), 367-380.  doi: 10.1086/655429. [36] J. N. Van Baalen, V. Krivan, P. C. J. Van Rijn and M. W. Sabelis, Alternative food, switching predators, and the persistence of predator-prey systems, Am. Nat., 157 (2001), 512-524.  doi: 10.1086/319933. [37] P. C. J. Van Rijn, Y. M. Van Houten and M. W. Sabelis, How plants benefit from providing food to predators even when it is also edible to herbivores, Ecology, 83 (2002), 2664-2679.
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 $1\leq\tau\leq3$
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
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