A Fractional-Order Food Chain Model with Omnivore and Anti-Predator

Authors

  • Adin Lazuardy Firdiansyah Department of PGMI, Faculty of Tarbiyah, State Islamic Institute of Madura, Pamekasan 69371, Indonesia

DOI:

https://doi.org/10.5614/cbms.2022.5.2.2

Keywords:

food chain, fractional-order, Grunwald-Letnikov approximation, stability

Abstract

A fractional-order food chain model is proposed in this article. The model is built by prey, intermediate predator, and omnivore. It is assumed that intermediate predator only eat prey and omnivore can consume prey and intermediate predator. But, prey has the ability called as anti-predator behavior to escape from both predators. For the first discussion, it is found that all solutions are existential, uniqueness, boundedness, and non-negative. Further, we analyze the existence condition and local stability of all points, that is point for the extinction of all populations, both predators, intermediate predator, omnivore, and point for the existence of all populations. We also investigate the global stability of all points, except point for the extinction of all populations and both predators. Finally, we preform several numerical solutions by using the nonstandard Grunwald-Letnikov approximation to demonstrate the our analytical results.

References

Li, Y., Chen, Y.Q. and Podlubny, I., Stability of Fractional-Order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag-Leffler Stability. Computers and Mathematics with Applications, 59(5), pp. 1810-1821, 2010.

Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 198, Academic Press, 1999.

Petras, I., Fractional-Order Nonlinear Systems: Modelling, Analysis, and Simulation, Springer Science and Business Media, 2011.

Namba, T., Tanabe, K., and Maeda, N., Omnivory and Stability of Food Webs, Ecological Complexity, 5(2), pp. 73-85, 2008.

Arim, M. and Marquet, P.A., Intraguild Predation: A Widespread Interaction Related to Species Biology, Ecology Letters, 7(7), pp. 557-564, 2004.

Jordan, F. and Scheuring, I., Network Ecology: Topological Constraints on Ecosystem Dynamics, Physics of Life Reviews, 1(3), pp. 139-172, 2004.

Tanabe, K. and Namba, T., Omnivory Creates Chaos in Simple Food Web Models, Ecology, 86(12), pp. 3411-3414, 2005.

Kang, Y. and Wedekin, L., Dynamics of a Intraguild Predation Model with Generalist or Specialist Predator, Journal of Mathematical Biology, 67(5), pp. 1227-1259, 2012.

Hsu, S., Ruan, S. and Yang, T., Analysis of Three Species Lotka-Volterra Food Web Models with Omnivory, Journal of Mathematical Analysis and Applications, 426(2), pp. 659-687, 2015.

Krikorian, N., The Volterra Model for Three Species Predator-Prey Systems: Boundedness and Stability, Journal of Mathematical Biology, 7(2), pp. 117-132, 1979.

Mccann, K. and Hastings, A., Re-Evaluating The Omnivory-Stability Relationship in Food Webs, In Proceedings of the Royal Society B: Biological Sciences, 264(1385), pp. 1249-1254, 1997.

Rosenheim, J.A. and Corbett, A., Omnivory and the Indeterminacy of Predator Function: Can a Knowledge of Foraging Behavior Help?, Ecology, 84(10), pp. 2538-2548, 2003.

Cai, Y., Zhao, C., Wang, W. and Wang, J., Dynamics of a Leslie-Gower Predator-Prey Model with Additive Allee Effect, Applied Mathematical Modelling, 39(7), pp. 2092-2106, 2015.

Holling, C.S., Some Characteristics of Simple Types of Predation and Parasitism, The Canadian Entomologist, 91(7), pp. 385-398, 1959.

Ahmed, E., El-Sayed, A. and El-Saka, H.A., On Some Routh-Hurwitz Conditions for Fractional Order Differential Equations and Their Applications in Lorenz, Rossler, Chua and Chen Systems, Physics Latters, 358(1), pp. 1-4, 2006.

Holt, R.D. and Polis, G.A., A Theoretical Framework for Intraguild Predation, The American Naturalist, 149(4), pp. 745-764, 1996.

Skalski, G.T. and Gilliam, J.F., Functional Responses with Predator Interference: Viable Alternatives to the Holling type II Model, Ecology, 82(11), pp. 3083-3092, 2001.

Vargas-De-Leon, C., Volterra-type Lyapunov Functions for Fractional-Order Epidemic Systems, Communications in Nonlinear Science and Numerical Simulation, 24(1-3), pp. 75-85, 2015.

Huo, J., Zhao, H. and Zhu, L., The Effect of Vaccines on Backward Bifurcation in a Fractional Order HIV Model, Nonlinear Analysis: Real World Applications, 26, pp. 289-305, 2015.

Choi, S.K., Kang, B. and Koo, N., Stability for Caputo Fractional Differential Systems, Abstract and Applied Analysis, 2014, pp. 1-6, 2014.

Wei, Z., Li, Q. and Che, J., Initial Value Problems for Fractional Differential Equations Involving Riemann-Liouville Sequential Fractional Derivative, Journal of Mathematical Analysis and Applications, 367(1), pp. 260-272, 2010.

Liu, J. and Zhang, L., Bifurcation Analysis in a Prey-Predator Model with Nonlinear Predator Harvesting, Journal of the Franklin Institute, 353(17), pp. 4701-4714, 2016.

Salamah, U., Suryanto, A. and Kusumawinahyu, W.M., Leslie-Gower Predator-Prey Model with Stage-Structure, Beddington-DeAngelis Functional Response, and Anti-Predator Behavior, In AIP Conference Proceedings, 2084(2019), p. 020001, 2018.

Panigoro, H.S., Resmawan, R., Sidik, A.T.R., Walangadi, N., Ismail, A., and Husuna, C., A Fractional-Order Predator-Prey Model with Age Structure on Predator and Nonlinear Harvesting on Prey, Jambura Journal of Mathematics, 4(2), pp. 355-366, 2022.

Panigoro, H.S., Suryanto, A., Kusumawinahyu, W.M., and Darti, I., A Rosenzweig-MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag-Leffler Kernel, Axioms, 9(122), pp. 1-22, 2020.

Panigoro, H.S., Suryanto, A., Kusumawinahyu, W.M., and Darti, I., Dynamics of an Eco-Epidemic Predator-Prey Model Involving Fractional Derivatives with Power-Law and Mittag-Leffler Kernel, Symmetry, 13(785), pp. 1-29, 2021.

Huda, M.N., Trisilowati and Suryanto, A., Dynamical Analysis of Fractional-Order Hastings-Powell Food Chain Model with Alternative Food, The Journal of Experimental Life Science, 7(1), pp. 39-44, 2017.

Sarwardi, S., Mandal, P.K. and Ray, S., Dynamical Behaviour of a Two-Predator Model with Prey Refuge, Journal of Biological Physics, 39(4), pp. 701-722, 2013.

Sayekti, I.M., Malik, M. and Aldila, D., One-prey Two-Predator Model with Prey Harvesting in a Food Chain Interaction, In AIP Conference Proceedings, 1862(1), p. 030124, 2017.

Sahoo, B. and Poria, S., Effects of Supplying Alternative Food in a Predator-Prey Model with Harvesting, Applied Mathematics and Computation, 234, pp. 150-166, 2014.

Sen, D., Ghorai, S. and Banerjee, M., Complex Dynamics of a Three Species Prey-Predator Model with Intraguild Predation, Ecological Complexity, 34, pp. 9-22, 2018.

Mukhopadhyay, B. and Bhattacharyya, R., Effects of Harvesting and Predator Interference in a Model of Two-predators Competing for a Single Prey, Applied Mathematical Modelling, 40(4), pp. 1-11, 2015.

Li, H.L., Zhang, L., Hu, C., Jiang, Y.L. and Teng, Z., Dynamical Analysis of a Fractional-Order Predator-Prey Model Incorporating a Prey Refuge, Journal of Applied Mathematics and Computing, 54(1-2), pp. 435-449, 2016.

Panigoro, H.S., Suryanto, A., Kusumawinahyu, W.M. and Darti, I., Dynamics of a Fractional-Order Predator-Prey Model with Infectious Diseases in Prey, Communication in Biomathematical Sciences, 2(2), pp. 105-117, 2019.

Nosrati, K. and Shafiee, M., Fractional-Order Singular Logistic Map: Stability, Bifurcation and Chaos Analysis, Chaos, Solitons and Fractals, 115, pp. 224-238, 2018.

Suryanto, A., Darti, I. and Anam, S., Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect, International Journal of Mathematics and Mathematical Sciences, 2017.

Magtinon, D., Stability Results for Fractional Differential Equations with Applications to Control Processing, In Computational engineering in systems applications, 2(1), pp. 963-968, 1996.

Nugraheni, K., Trisilowati and Suryanto, A., Dynamics of a Fractional Order Eco-Epidemiological Model, Journal of Tropical Life Science, 7(3), pp. 243-250, 2017.

Rahmi, E., Darti, I., Suryanto, A., Trisilowati and Panigoro, H.S., Stability Analysis of a Fractional-Order Leslie-Gower Model with Allee Effect in Predator, In Journal of Physics Conference Series, 1821(1), p. 012051. 2021.

Scherer, R., Kalla, S.L., Tang, Y. and Huang, J., The Grunwald-Letnikov Method for Fractional Differential Equations, Computers and Mathematics with Applications, 62(3), pp. 902-917, 2011.

Arenas, A.J., Gonzalez-Parra. G. and Chen-Charpentier. B.M., Construction of Nonstandard Finite Difference Schemes for the SI and SIR Epidemic Models of Fractional Order, Mathematics and Computers in Simulation, 121, pp. 48-63, 2016.

Mickens, R.E., Nonstandard Finite Difference Models of Differential Equations, World Scientific Publishing, 1994.

Mickens, R.E., Applications of Nonstandard Finite Difference Schemes, World Scientific Publishing, 1999.

Mickens, R.E., Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific Publishing, 2005.

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Published

2023-01-03

How to Cite

Firdiansyah, A. L. (2023). A Fractional-Order Food Chain Model with Omnivore and Anti-Predator. Communication in Biomathematical Sciences, 5(2), 121-136. https://doi.org/10.5614/cbms.2022.5.2.2

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