# Analysis of Prey, Predator and Top Predator Model Involving Various Functional Responses

## Keywords:

Food chain model, Functional Response, Global Stability, Jacobian Matrix, Logistic Growth, Stability Analysis

## Abstract

This research proposes a mathematical model to investigate the dynamical behavior of the system of three species, namely prey, predator and top predator. The feeding behavior of each predator serves as a functional response. The interaction between the species is carried out by a functional response. Crowley Martin functional response is incorporated between prey and predator while Holling type III functional response occurs between predator and top predator­. The existence of positivity and boundedness of the system have been examined. The equilibrium points of the system are determined. The system has been linearized by applying the Jacobian matrix. The main perspective used to discuss the system's dynamics is that of permanence and stability. Further stability analysis of the system is carried out at around each equilibrium point. To comprehend the dynamics of the model system, the asymptotic stability of several equilibrium solutions, both local and global, is investigated. Routh Hurwitz criteria are used to analyze local stability at every equilibrium point. Using an appropriate Lyapunov function, the global asymptotic stability of the positive interior equilibrium solution is established. From a biological perspective, a system is considered to be permanent if all of its populations continue to exist in the future. The existence of permanence conditions of the system have been determined. To support the analytical results, several numerical simulations are carried out using the MATLAB software. Finally based on the results of the analytical and numerical simulations, the impact of the functional response between the prey, predator and top predator was discussed.

## References

Kapur JN. Mathematical Models in Biology and Medicines. India: Affiliated East-West Press; 1985. 520 p.

Kapur JN. Mathematical Modelling. India: John Wiley & Sons; 1988. 259 p.

Murray JD. Mathematical biology: I. An introduction. 3rd edition. Interdisciplinary Applied Mathematics (IAM) New York: Springer; 2002; 17: 551 p. https://doi.org/10.1007/b98868.

Holling CS. The Components of Predation as Revealed by a Study of Small-Mammal Predation of the European Pine Sawfly. Can Entomol. 1959 May; 91(5): 293-320. https://doi.org/10.4039/Ent91293-5

Molla H, Sarwardi S, Sajid M. Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge. J Math Comput Sci. 2022 April 20; 25(2):150-165. http://dx.doi.org/10.22436/jmcs.025.02.04

Sharmila NB, Gunasundari C, Sajid M. Spatiotemporal Dynamics of a Reaction Diffusive Predator-Prey Model: A Weak Nonlinear Analysis. Int J Differ Equ. 2023 Oct; 2023: 1-23. https://doi.org/10.1155/2023/9190167

Kerner EH. On the Volterra-Lotka principle. Bull Math Biophys. 1961 Jun; 23: 141-157. https://doi.org/10.1007/BF02477468

Chauvet E, Paullet JE, Previte JP, Walls Z. A Lotka-Volterra three-species food chain. Math Mag. 2002 Oct; 75(4): 243-255. https://doi.org/10.1080/0025570X.2002.11953139

Hastings A, Powell T. Chaos in a Three-Species Food Chain. Ecology. 1991 Jun; 72(3): 896-903. https://doi.org/10.2307/1940591

Boudjellaba H, Sari T. Oscillations in a Prey-Predator-Superpredator System. J Biol Syst. 1998 Mar; 6(1): 17-33. https://doi.org/10.1142/S0218339098000066

Klebanoff A, Hastings A. Chaos in three species food chains. J Math Biol. 1994 May; 32: 427-451. https://doi.org/10.1007/BF00160167

Do Y, Baek H, Lim Y, Lim D. A Three-Species Food Chain System with Two Types of Functional Responses. Abstr Appl Anal. 2011 Jan; 2011: 1-16. https://doi.org/10.1155/2011/934569

Upadhyay RK, Naji RK. Dynamics of a three species food chain model with Crowley–Martin type functional response. Chaos Solit Fractals. 2009 Nov; 42(3): 1337-1346. https://doi.org/10.1016/j.chaos.2009.03.020

Panja P. Stability and dynamics of a fractional-order three-species predator–prey model. Theory Biosci. 2019 Nov; 138: 251-259. https://doi.org/10.1007/s12064-019-00291-5

Sunaryo MS, Salleh Z, Mamat M. Mathematical Model of Three Species Food Chain with Holling Type-III Functional Response. Int J Pure Appl Math. 2013; 89(5): 647-657. http://dx.doi.org/10.12732/ijpam.v89i5.1

Krishnadas M, Saratchandran PP, Harikrishnan KP. Chaos in a cyclic three-species predator–prey system with a partial consumption of superpredator. Pramana. 2020 Dec; 94: 1-2. https://doi.org/10.1007/s12043-020-1942-9

Khan AQ, Qureshi SM, Alotaibi AM. Bifurcation analysis of a three species discrete-time predator-prey model. Alex Eng J. 2022 Oct; 61(10): 7853-75. https://doi.org/10.1016/j.aej.2021.12.068

Jana A, Roy SK. Fostering roles of super predator in a three-species food chain. Int J Dynam Control. 2023 Feb; 11(1): 78-93. http://dx.doi.org/10.1007/s40435-022-00970-0

Ghosh U, Sarkar S, Mondal B. Study of Stability and Bifurcation of Three Species Food Chain Model with Non-monotone Functional Response. Int J Appl Comput Math. 2021 Jun; 7: 1-24. https://doi.org/10.1007/s40819-021-01017-2

Mondal A, Pal AK, Samanta GP. Stability and Bifurcation Analysis of a Delayed Three Species Food Chain Model with Crowley-Martin Response Function. Appl Appl Math. Int J. 2018 Dec; 13(2): 709-749.

Goodman D. The Theory of Diversity-Stability Relationships in Ecology. Q Rev Biol. 1975 Sep; 50(3): 237-266. https://doi.org/10.1086/408563

Gard TC, Hallam TG. Persistence in food webs—I Lotka-Volterra food chains. Bull Math Biol. 1979 Jan; 41(6): 877-91. https://doi.org/10.1007/BF02462384.

Gard TC. Persistence in food chains with general interactions. Math Biosci. 1980 Sep; 51(1-2): 165-174. https://doi.org/10.1016/0025-5564(80)90096-6

Arif GE, Alebraheem J, Yahia WB. Dynamics of Predator-prey Model under Fluctuation Rescue Effect. Baghdad Sci J. 2023 Feb; 20(5): 1742-1750. https://doi.org/10.21123/bsj.2023.6938

Ali SJ, Atewi AN, Naji RK. The Complex Dynamics in a Food Chain Involving Different Functional Responses. Iraqi J Sci. 2022 Apr; 63(4): 1747-54. https://doi.org/10.24996/ijs.2022.63.4.32

Naji RK. On The Dynamical Behavior of a Prey-Predator Model With The Effect of Periodic Forcing. Baghdad Sci J. 2021 Mar. 10; 4(1): 147-157.

Ali SJ, Arifin NM, Naji RK, Ismail F, Bachok N. Global Stability of a Three Species Predator-Prey Food Chain Dynamics. Dyn Contin. Discrete Impuls Syst B Appl Algorithms. 2019; 26(1b):39-52.

Ali SJ, Almohasin AA, Atewi AN, Naji RK, Arifin NM. Chaos in a Hybrid Food Chain Model. Iraqi J Sci. 2021 Jul; 62(7): 2362-2368. https://doi.org/10.24996/ijs.2021.62.7.25

Chen M, Zheng Q. Global Stability of a Three-Species System with Attractive Prey-Taxis. Appl Math. 2022 Aug; 13(08): 658-71. https://doi.org/10.4236/am.2022.138041

Arumugam G. Global existence and stability of three species predator-prey system with prey-taxis. Math Biosci Eng. 2023; 20(5): 8448-8475. https://doi.org/10.3934/mbe.2023371

Das K, Srinivash MN, Kabir MH, Gani MO. Noise-induced control of environmental fluctuations in a three-species predator–prey model. Model. Earth Syst Environ. 2021 Nov; 7: 2675-2695. https://doi.org/10.1007/s40808-020-01051-x

Danane J, Torres DF. Three-Species Predator–Prey Stochastic Delayed Model Driven by Lévy Jumps and with Cooperation among Prey Species. Mathematics. 2023 Mar 25; 11(7): 1595. https://doi.org/10.3390/math11071595

Ikbal M. Dynamics of Predator-Prey Model Interaction with Intraspecific Competition. 4th International Conference on Mathematics, Science, Education and Technology (ICOMSET) in Conjunction with the 2nd International Conference on Biology, Science and Education (ICoBioSE) 2020 23-24 July 2020, Padang, Indonesia. J Phys Conf Ser. 2021 Jun; 1940(1): 1-8. https://doi.org/10.1088/1742-6596/1940/1/012006

Manaqib M, Zahra A. Mathematical Model of Three Species Food Chain with Intraspecific Competition and Harvesting on Predator. Barekeng: J Mat App 2022 Jun; 16(2): 551-562. https://doi.org/10.30598/barekengvol16iss2pp551-562

Liu R, Liu G. Dynamics of a Stochastic Three Species Prey-Predator Model with Intraguild Predation. J Appl Anal Comput. 2020 Feb; 10(1): 81-103. https://doi.org/10.11948/jaac20190002

Ghosh P, Das P, Mukherjee D. Persistence and Stability of a Seasonally Perturbed Three Species Stochastic Model of Salmonoid Aquaculture. Differ Equ Dyn Syst. 2019 Oct; 27: 449-465. https://doi.org/10.1007/s12591-016-0283-0

Zhou D, Liu M, Liu Z. Persistence and extinction of a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. Adv Differ Equ. 2020 Dec; 2020(1): 1-5. https://doi.org/10.1186/s13662-020-02642-9