Analysis Dynamics Two Prey of a Predator-Prey Model with Crowley–Martin Response Function

Rian Ade Pratama; Syamsuddin Toaha

doi:10.31764/jtam.v7i3.14506

Analysis Dynamics Two Prey of a Predator-Prey Model with Crowley–Martin Response Function

Rian Ade Pratama, Syamsuddin Toaha

Abstract

The predator-prey model has been extensively developed in recent research. This research is an applied literature study with a proposed dynamics model using the Crowley–Martin response function, namely the development of the Beddington-DeAngelis response function. The aim of this research is to construct a mathematical model of the predator-prey model, equilibrium analysis and population trajectories analysis. The results showed that the predator-prey model produced seven non-negative equilibrium points, but only one equilibrium point was tested for stability. Stability analysis produces negative eigenvalues indicating fulfillment of the Routh-Hurwitz criteria so that the equilibrium point is locally asymtotically stable. Analysis of the stability of the equilibrium point indicates stable population growth over a long period of time. Numerical simulation is also given to see the trajectories of the population growth movement. The population growth of first prey and second prey is not much different, significant growth occurs at the beginning of the growth period, while after reaching the peak the trajectory growth slopes towards a stable point. Different growth is shown by the predator population, which grows linearly with time. The growth of predators is very significant because predators have the freedom to eat resources. Various types of trajectory patterns in ecological parameters show good results for population growth with the given assumptions.

Keywords

Dynamics; Predator-Prey; Crowley–Martin; Population Modeling.

Full Text:

DOWNLOAD [PDF]

References

Batabyal, S., Jana, D., Lyu, J., & Parshad, R. D. (2020). Explosive predator and mutualistic preys: A comparative study. Physica A: Statistical Mechanics and Its Applications, 541. https://doi.org/10.1016/j.physa.2019.123348

Ghanbari, B., Günerhan, H., & Srivastava, H. M. (2020). An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model. Chaos, Solitons and Fractals, 138, 109910. https://doi.org/10.1016/j.chaos.2020.109910

Hossain, S., Haque, M. M., Kabir, M. H., Gani, M. O., & Sarwardi, S. (2021). Complex spatiotemporal dynamics of a harvested prey–predator model with Crowley–Martin response function. Results in Control and Optimization, 5(July), 100059. https://doi.org/10.1016/j.rico.2021.100059

Kulkarni, R. G. (2008). Solving Sextic Equations. Journal of Mathematics, 3(1), 56–60. http://euclid.trentu.ca/aejm/V3N1/Kulkarni.V3N1.pdf

Maiti, A. P., Dubey, B., & Chakraborty, A. (2019). Global analysis of a delayed stage structure prey–predator model with Crowley–Martin type functional response. Mathematics and Computers in Simulation, 162, 58–84. https://doi.org/10.1016/j.matcom.2019.01.009

May, R. M. (1973). Stability and complexity in model ecosystems. Monographs in Population Biology, 6, 1–235. https://doi.org/10.2307/3743

Meng, X. Y., Huo, H. F., Xiang, H., & Yin, Q. Y. (2014). Stability in a predator-prey model with Crowley-Martin function and stage structure for prey. Applied Mathematics and Computation, 232, 810–819. https://doi.org/10.1016/j.amc.2014.01.139

Mortoja, S. G., Panja, P., & Mondal, S. K. (2019). Dynamics of a predator-prey model with nonlinear incidence rate, Crowley-Martin type functional response and disease in prey population. Ecological Genetics and Genomics, 10, 100035. https://doi.org/10.1016/j.egg.2018.100035

Parshad, R. D., Basheer, A., Jana, D., & Tripathi, J. P. (2017). Do prey handling predators really matter: Subtle effects of a Crowley-Martin functional response. Chaos, Solitons and Fractals, 103, 410–421. https://doi.org/10.1016/j.chaos.2017.06.027

Pratama, R. A. (2022). IMPACT OF FEAR BEHAVIOR ON PREY POPULATION GROWTH PREY-PREDATOR INTERACTION. BAREKENG: Jurnal Ilmu Matematika Dan Terapan, 16(2), 371–378. https://doi.org/10.30598/barekengvol16iss2pp371-378%0AIMPACT

Pratama, R. A., Dadi, O., Siddik, A. M. A., & Kasbawati. (2022). Hydra effects predator-prey bazykin’s model with stage- structure and intraspecific for predator. DESIMAL: JURNAL MATEMATIKA, 5(3), 279–288. https://doi.org/10.24042/djm

Pratama, R. A., Toaha, S., & Kasbawati. (2019). Optimal harvesting and stability of predator prey model with Monod-Haldane predation response function and stage structure for predator. IOP Conference Series: Earth and Environmental Science, 279(1), 0–7. https://doi.org/10.1088/1755-1315/279/1/012015

Puspitasari, N., Kusumawinahyu, W. M., & Trisilowati, T. (2021). Dynamic Analysis of the Symbiotic Model of Commensalism and Parasitism with Harvesting in Commensal Populations. JTAM (Jurnal Teori Dan Aplikasi Matematika), 5(1), 193. https://doi.org/10.31764/jtam.v5i1.3893

Shang, Z., Qiao, Y., Duan, L., & Miao, J. (2021). Bifurcation analysis in a predator–prey system with an increasing functional response and constant-yield prey harvesting. Mathematics and Computers in Simulation, 190, 976–1002. https://doi.org/10.1016/j.matcom.2021.06.024

Tripathi, J. P., Bugalia, S., Tiwari, V., & Kang, Y. (2020). A predator–prey model with Crowley–Martin functional response: A nonautonomous study. Natural Resource Modeling, 33(4), 1–49. https://doi.org/10.1111/nrm.12287

Tripathi, J. P., Tyagi, S., & Abbas, S. (2016). Global analysis of a delayed density dependent predator-prey model with Crowley-Martin functional response. Communications in Nonlinear Science and Numerical Simulation, 30(1–3), 45–69. https://doi.org/10.1016/j.cnsns.2015.06.008

Yang, R. (2017). Bifurcation analysis of a diffusive predator–prey system with Crowley–Martin functional response and delay. Chaos, Solitons and Fractals, 95, 131–139. https://doi.org/10.1016/j.chaos.2016.12.014

Yin, H., Xiao, X., Wen, X., & Liu, K. (2014). Pattern analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional response and diffusion. Computers and Mathematics with Applications, 67(8), 1607–1621. https://doi.org/10.1016/j.camwa.2014.02.016

Yulida, Y., & Karim, M. A. (2019). Analisa Kestabilan dan Solusi Pendekatan Pada Persamaan Van der Pol. JTAM | Jurnal Teori Dan Aplikasi Matematika, 3(2), 156. https://doi.org/10.31764/jtam.v3i2.1084

DOI: https://doi.org/10.31764/jtam.v7i3.14506