Dynamical Behavior in Prey-Predator Model with Mutualistic Protection for Prey

Laras Kinanti Putri, Dian Savitri, Abadi Abadi


This article reconstructs the model of predator-prey mutualistic protection based on a journal written by Revilla and Krivan (2022). The predator-prey model considers mutualistic protection for the prey. The model focuses on the analysis of equilibrium points and combines an adaptive model to study the influence of both models on predator-prey dynamics. This research continues the stability analysis and numerical simulations of the predator-prey model with mutualistic protection to examine the impact of mutualistic protection on prey dynamics in the model. The research process begins with a literature review, reconstructing the predator-prey model, determining equilibrium points, analyzing stability at the equilibrium points, conducting numerical simulations including bifurcation diagrams and phase portraits of the model solutions, and drawing conclusions. The analysis yields three equilibrium points: the unstable co-extinction of both populations, predator extinction, and the conditionally stable coexistence of both populations. Based on the analysis results, there are changes in the system solutions, with the originally stable E3 becoming unstable. There is also a change in E2 from being unstable to stable. Through numerical continuation with variations in the parameter representing the mutualistic protector’s preference for prey resources (u), a transcritical bifurcation (Branch Point) is obtained at u = 0.888889. The simulation results demonstrate that (u) can influence the stability of predator and prey populations.


Protection Mutualism; Stability; Numerical Simulation

Full Text:



D. Savitri and H. S. Panigoro, “Bifurkasi Hopf pada model prey-predator-super predator dengan fungsi respon yang berbeda,” Jambura Journal of Biomathematics (JJBM), vol. 1, no. 2, pp. 65–70, 2020. DOI:10.34312/jjbm.v1i2.8399

D. Mukherjee, “Stability and bifurcation of a two competing prey-one predator system with anti-predator behavior,” Jambura Journal of Biomathematics (JJBM), vol. 3, no. 1, pp. 1–11, 2022. DOI:10.34312/jjbm.v3i1.13820

A. L. Firdiansyah and N. Nurhidayati, “Dynamics in two competing predators-one prey system with two types of Holling and fear effect,” Jambura Journal of Biomathematics (JJBM), vol. 2, no. 2, pp. 58–67, 2021. DOI:10.34312/jjbm.v2i2.11264

H. S. Panigoro and E. Rahmi, “Computational dynamics of a Lotka-Volterra Model with additive Allee effect based on Atangana-Baleanu fractional derivative,” Jambura Journal of Biomathematics (JJBM), vol. 2, no. 2, pp. 96–103, 2021. DOI:10.34312/jjbm.v2i2.11886

X. Xue, G. Li, D. Zhou, Y. Zhang, L. Zhang, Y. Zhao, Z. Feng, L. Cui, Z. Zhou, X. Sun, X. Lu, and S. Chen, “Research Roadmap of Service Ecosystems: A Crowd Intelligence Perspective,” International Journal of Crowd Science, vol. 6, no. 4, pp. 195–222, 2022. DOI:10.26599/IJCS.2022.9100026

C. Hui, D. M. Richardson, P. Landi, H. O. Minoarivelo, H. E. Roy, G. Latombe, X. Jing, P. J. CaraDonna, D. Gravel, B. Beckage, and J. Molofsky, “Trait Positions for Elevated Invasiveness in Adaptive Ecological Networks,” Biological Invasions, vol. 23, no. 6, pp. 1965–1985, 2021. DOI:10.1007/s10530-021-02484-w

R. E. Irwin, E. Youngsteadt, P. S. Warren, and J. Bronstein, “The evolutionary ecology of mutualisms in urban landscapes,” Urban evolutionary biology, pp. 112–129, 2020.

S. Rajashekara, S. S. Devi, and M. Venkatesha, Biotechnological Tools for Monitoring, Assessment, and Insect Pest Management in Agricultural Ecosystems. Springer, 2022, pp. 315–390. ISBN 978-3-030-94948-8

S. Reddy, K. Kaur, P. Barathe, V. Shriram, M. Govarthanan, and V. Kumar, “Antimicrobial resistance in urban river ecosystems,” Microbiological Research, vol. 263, p. 127135, 2022. DOI:10.1016/j.micres.2022.127135

A. Yamawo, “Intraspecific competition favors ant–plant protective mutualism,” Plant Species Biology, vol. 36, no. 3, pp. 372–378, 2021. DOI:10.1111/1442-1984.12331

E. A. Rocha and M. D. Fellowes, “Urbanisation Alters Ecological Interactions: Ant Mutualists Increase and Specialist Insect Predators Decrease On an Urban Gradient,” Scientific Reports, vol. 10, no. 1, pp. 1–8, 2020. DOI:10.1038/s41598-020-62422-z

M. Hullé, B. Chaubet, E. Turpeau, and J.-C. Simon, “Encyclop’aphid: a website on aphids and their natural enemies,” Entomologia Generalis, vol. 40, no. 1, pp. 97–101, 2020. DOI:10.1127/entomologia/2019/0867

P. A. Ortega-Ramos, E. T. Mezquida, and P. Acebes, “Ants Indirectly Reduce the Reproductive Performance of a Leafless Shrub by Benefiting Aphids Through Predator Deterrence,” Plant Ecology, vol. 221, no. 2, pp. 91–101, 2020. DOI:10.1007/s11258-019-00995-0

T. A. Revilla and V. Krˇivan, “Prey–predator dynamics with adaptive protection mutualism,” Applied Mathematics and Computation, vol. 433, p. 127368, 2022. DOI:10.1016/j.amc.2022.127368

G. Gabbriellini, “Aphids, ants and ladybirds: a mathematical model predicting their population dynamics,” Open Journal of Mathematical Sciences, vol. 3(2019), pp. 139–151, 2019. DOI:10.30538/oms2019.0057

S. Swire, E. Pasipanodya, M. A. Morales, and E. Peacock-López, “Complex dynamics in a minimal model of protection-based mutualism,” Axioms, vol. 9, no. 1, p. 26, 2020. DOI:10.3390/axioms9010026

J. Samuel and N. Rastogi, “Consumptive and non-consumptive effects of a generalist and a specialist arthropod predator on ant-tended aphids,” International Journal of Zoological Research, vol. 7, no. 2, pp. 433–439, 2021. DOI:10.33745/ijzi.2021.v07i02.016

J. Samuel and N. Rastogi, “Do ant-mediated multitrophic interactions enhance the fitness of an aphid-infested extrafloral nectary-bearing plant?” Proceedings of the National Academy of Sciences, India – Section B: Biological Sciences, vol. 92, pp. 1–8, 2022. DOI:10.1007/s40011-021-01312-4

D. Korányi, V. Szigeti, L. Mezfi, E. Kondorosy, and V. Markó, “Urbanization alters the abundance and composition of predator communities and leads to aphid outbreaks on urban trees,” Urban Ecosystems, vol. 24, pp. 571–586, 2021. DOI:10.1007/s11252-020-01061-8

A. S. Nelson, R. T. Pratt, J. D. Pratt, R. A. Smith, C. T. Symanski, C. Prenot, and K. A. Mooney, “Progressive sensitivity of trophic levels to warming underlies an elevational gradient in ant–aphid mutualism strength,” Oikos, vol. 128, no. 4, pp. 540–550, 2019. DOI:10.1111/oik.05650

B. Wang, M. Lu, Y.-Q. Peng, and S. T. Segar, “Direct and indirect effects of invasive vs. native ant-hemipteran mutualism: A meta–analysis that supports the mutualism intensity hypothesis,” Agronomy, vol. 11, no. 11, p. 2323, 2021. DOI:10.3390/agronomy11112323

L. Leal, A. Nogueira, and P. E. Peixoto, “Which traits optimize plant benefits? meta-analysis on the effect of partner traits on the outcome of an ant-plant protective mutualism,” Journal of Ecology, vol. 111, no. 1, pp. 263–275, 2023. DOI:10.1111/1365-2745.14031

K. R. Hale, D. P. Maes, and F. S. Valdovinos, “Ecological theory of mutualism: Models generalizing across different mechanisms,” bioRxiv, pp. 2020–10, 2020. DOI:10.1101/2020.10.25.343087

K. R. S. Hale and F. S. Valdovinos, “Ecological theory of mutualism: Robust patterns of stability and thresholds in two–species population models,” Ecology and Evolution, vol. 11, no. 24, pp. 17 651–17 671, 2021. DOI:10.1002/ece3.8453

DOI: https://doi.org/10.37905/jjbm.v4i2.21541

Copyright (c) 2023 Laras Kinanti Putri, Dian Savitri, Abadi Abadi

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Jambura Journal of Biomathematics (JJBM) has been indexed by:


 Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Gorontalo
Jl. Prof. Dr. Ing. B. J. Habibie, Moutong, Tilongkabila, Kabupaten Bone Bolango 96554, Gorontalo, Indonesia
 Email: editorial.jjbm@ung.ac.id
 +6281356190818 (Call/SMS/WA)
 Jambura Journal of Biomathematics (JJBM) by Department of Mathematics Universitas Negeri Gorontalo is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.  Powered by Public Knowledge Project OJS.