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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi
Dian Savitri
Department of Mathematics, Faculty of Mathematics and Natural Sciences, State University of Surabaya, Surabaya
Mathematics

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Stability Analysis and Invasion Thresholds in a Rosenzweig--MacArthur Model with Prey Immigration and Cooperative Hunting Naufal Daffa Faustin; Dian Savitri
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i1.40677

Abstract

Predator--prey systems with multiple interacting ecological mechanisms require integrated modeling approaches for realistic analysis. This study develops a unified Rosenzweig--MacArthur model incorporating both continuous prey immigration and cooperative hunting among predators to analyze how these combined mechanisms affect equilibrium existence, stability, and transient dynamics. Analytical methods derive explicit invasion thresholds and local stability conditions through eigenvalue analysis, while numerical simulations with biologically plausible parameters compare two dynamical regimes: baseline conditions produce stable-node convergence, whereas high-efficiency conditions yield stable-spiral oscillations. Results show that immigration elevates prey density above invasion thresholds, enabling predator persistence, while increased cooperation intensity transitions the system from monotonic to oscillatory convergence. The integrated framework demonstrates how bottom-up (immigration) and top-down (cooperation) processes interact to shape predator--prey dynamics, providing testable predictions for ecosystems where both mechanisms operate simultaneously and establishing a foundation for more complex ecological modeling.
Stability Analysis of a Three-Trophic Rice Ecosystem with Holling Type II and Crowley--Martin Responses Anastasya Choirun Nisa'; Dian Savitri
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i1.40204

Abstract

This study examines three-trophic-level ecological dynamics in rice field ecosystems involving rice as the producer, brown planthoppers Nilaparvata lugens as herbivores, and Pardosa sp. as top predators. Population interactions were modeled using logistic growth for rice and two response functions: the Holling type II response for planthopper predation on rice, and the Crowley--Martin response for Pardosa sp. predation on planthoppers to capture predator interference. The resulting model was formulated as a system of nonlinear differential equations. Analytically, four equilibrium points were obtained: total population extinction, extinction of planthoppers and Pardosa sp., extinction of Pardosa sp., and coexistence of all three species. Local stability analysis at each equilibrium point was conducted using the Jacobian matrix and the Routh--Hurwitz criterion. Numerical simulations were performed for several parameter combinations by varying planthopper growth efficiency and Pardosa sp. predation intensity. Parameter values are obtained from relevant literature where available, and the remaining parameters are considered reasonable biological assumptions for exploring the dynamics of the system qualitatively. The simulation results were consistent with the theoretical analysis and showed that small changes in biological parameters could shift the system from stable coexistence to near extinction of the planthopper population. Ecologically, the model demonstrates that rice field ecosystem balance is strongly influenced by the interaction between planthopper reproductive capacity and the predatory strength of Pardosa sp., providing theoretical insights that may support the development of sustainable planthopper pest management strategies.
Stage-Structured Predator Model with Prey Protection: Application to Rice Plants–Leptocorisa oratorius Safira Rahmah; Dian Savitri
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i1.40733

Abstract

This study investigates a stage-structured predator–prey model consisting of prey, juvenile predators, and adult predators. The prey population follows logistic growth, while predation is described using a Holling type I functional response. Prey protection is incorporated through a protection parameter (1-m), representing the proportion of prey that successfully avoid predation by reducing the predation rate of adult predators. The model is analyzed by determining equilibrium points and examining their existence and stability. The results show four equilibrium points: total population extinction, prey-only equilibrium, juvenile predator extinction, and coexistence equilibrium. Predator extinction occurs when predation efficiency and predator reproduction are insufficient to compensate for predator mortality, whereas coexistence occurs when predation and conversion rates exceed mortality thresholds. Numerical simulations support the analytical results and demonstrate that increasing prey protection reduces predation pressure and may lead to predator decline, while appropriate predation efficiency promotes stable coexistence. These findings highlight the ecological importance of prey defense mechanisms in predator–prey interactions, particularly in rice–Leptocorisa oratorius.
Dynamic Analysis of a Predator–Prey Model with Group Defense in Prey and Cooperative Hunting in Predators Nabilah Meladelvia; Dian Savitri
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i1.40210

Abstract

This article discusses a single-prey and single-predator model by incorporating two behavioral mechanisms, namely group defense in prey modeled through a Holling type IV response function and cooperative hunting in predators represented by a predation rate dependent on predator density. Through system analysis, up to four equilibrium points are obtained mathematically. Among these, three equilibria are biologically feasible under typical parameter values, corresponding to total extinction, predator extinction, and coexistence states. The total extinction equilibrium is always unstable, while the stability of the predator extinction and coexistence equilibria depends on the predator attack rate. Numerical simulations in the form of phase portraits were obtained by varying the parameters related to the predator attack rate. The simulation results show various dynamic behaviors, including predator extinction, asymptotically stable coexistence between prey and predators, and bistability. Numerical continuation analysis identifies a subcritical Hopf bifurcation at α=0.4595, confirmed by a positive first Lyapunov coefficient, as well as a saddle-node at α= 0.0478 and transcritical bifurcations α=3.0505 that alter equilibrium structure and stability. These findings demonstrate how prey group defense and predator cooperation can generate bistability and abrupt transitions between extinction and coexistence, accompanied by damped oscillatory dynamics near critical parameter values.
Co-Authors Anastasya Choirun Nisa' Nabilah Meladelvia Naufal Daffa Faustin Safira Rahmah