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All Journal MUSTEK ANIM HA Desimal: Jurnal Matematika BAREKENG: Jurnal Ilmu Matematika dan Terapan JTAM (Jurnal Teori dan Aplikasi Matematika) MATAPPA: Jurnal Pengabdian Kepada Masyarakat JURNAL SILOGISME : Kajian Ilmu Matematika dan Pembelajarannya JPM : Jurnal Pendidikan Matematika Klorofil: Jurnal Penelitian Ilmu-Ilmu Pertanian MATHEMA: JURNAL PENDIDIKAN MATEMATIKA Jurnal Inovasi Pembelajaran Matematika (JIPM)
Pratama, Rian Ade
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Agriculture, Biological Sciences & Forestry Computer Science & IT Control & Systems Engineering Economics, Econometrics & Finance Education Electrical & Electronics Engineering Energy Engineering Mathematics Mechanical Engineering Physics Social Sciences Transportation

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Journal : JTAM (Jurnal Teori dan Aplikasi Matematika)

 1 
Analysis Dynamics Model Predator-Prey with Holling Type III Response Function and Anti-Predator Behavior Pratama, Rian Ade; Suryani, Dessy Rizki; Ruslau, Maria F. V.; Meirista, Etriana; Nurhayati, Nurhayati
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 9, No 3 (2025): July
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v9i3.31533

Abstract

Understanding predator-prey dynamics is essential for maintaining ecological balance and biodiversity. Classical models often fail to capture complex biological behaviors such as prey defense mechanisms and nonlinear predation effects, which are vital for accurately describing real ecosystems. In light of this, there is a growing need to incorporate behavioral and functional complexity into mathematical models to better understand species interactions and their long-term ecological outcomes. This study aims to develop and analyze a predator-prey model that integrates two key ecological features: a Holling type III functional response and the anti-predator behavior exhibited by prey. The model assumes a habitat with limited carrying capacity to reflect environmental constraints. We formulate a nonlinear system of differential equations representing the interaction between prey and predator populations. The model is examined analytically by identifying equilibrium points and analyzing their local stability using the Routh-Hurwitz criteria. A literature-based theoretical analysis is supplemented with numerical simulations to validate and illustrate population dynamics. The model exhibits three equilibrium points: a trivial solution (extinction), a predator-free equilibrium, and a non-trivial saddle point representing coexistence. The non-trivial equilibrium best reflects ecological reality, indicating stable coexistence where prey consumption is balanced by reproduction, and predator mortality aligns with energy intake. Numerical simulations show that prey populations initially grow rapidly, then decline as they reach carrying capacity, while predator populations grow after a time lag and eventually stabilize. The results are further supported by the eigenvalue analysis, confirming local asymptotic stability. The proposed model realistically captures predator-prey dynamics, demonstrating that the inclusion of anti-predator behavior and a Holling type III response significantly affects population trajectories and system stability. This framework provides a more ecologically valid approach for studying long-term species coexistence and highlights the importance of incorporating behavioral responses in ecological modeling.
Modeling Predator-Prey Interactions Barramundi in Dogamit Swamp Wasur National Park Merauke Pratama, Rian Ade; Ruslau, Maria F V; Suryani, Dessy Rizki; Nurhayati, Nurhayati; Meirista, Etriana
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 8, No 4 (2024): October
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v8i4.25563

Abstract

Dogamit, which serves as a habitat for fish species growth, has drawn attention due to its location within a national park and the practice of 'sasi' by the local community as a way to preserve the ecosystem and the species that interact within it. In this research, mathematical modeling variables are explained to describe species' life based on direct observation. As the ecosystem’s inhabitants, the dominant predator species in the ecosystem is the Barramundi fish. Historically, this predator species has migrated from the waters of Australia. The aim of this research is to determine the locally stable equilibrium point and analyze the growth trajectories of the species. The testing is conducted based on equilibrium point analysis. There are three equilibrium points, but only one is a non-negative and realistic point for stability testing. This equilibrium point is then tested using the Routh-Hurwitz criteria. Stability is analyzed using the Jacobian matrix to obtain the eigenvalues. All eigenvalues are negative, thus it can be concluded that the model tested is locally stable. A numerical simulation analysis is also provided, involving parameters that support the mathematical model. The parameters are derived from previous relevant studies and realistic assumptions. The numerical simulation analysis method is used to observe the population growth trajectories. The trajectories that appear show similar conditions for both populations. Both populations experience significant fluctuations with an average growth rate of 67%. It takes 3/5 of the species' lifespan for both populations to stabilize again within the ecosystem. The predator-prey populations also demonstrate resilience during fluctuations, indicating that both populations are highly robust in maintaining survival. The characteristics and findings of this research are commonly found only in endemic species populations. Endemic species tend to have long-term survival and endurance, allowing them to dominate their surrounding geographic habitat and maintain ecosystem balance. 
Dynamical of Prey Refuge in a Diased Predator-Prey Model with Intraspecific Competition for Predator Pratama, Rian Ade
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 8, No 2 (2024): April
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v8i2.19893

Abstract

The predator-prey model described is a population growth model of an eco-epidemiological system with prey protection and predator intraspecific traits. Predation interactions in predator species use response functions. The aim of this research is to examine the local stable balance point and look at the characteristics of species resulting from mathematical modeling interventions. Review of balance point analysis, numerical simulation and analysis of given trajectories. The research results show the shape of the model which is arranged with a composition of 5 balance points. There is one rational balance point to be explained, using the Routh-Hurwitz criterion, . The characteristic equation and associated eigenvalues in the mathematical model are the local asymptotically stable balance points. In the trajectory analysis, local stability is also shown by the model formed. There are differences for each population to reach its point of stability. The role of prey protection behavior is very effective in suppressing the spread of disease. Meanwhile, intraspecific predator interactions are able to balance the decreasing growth of prey populations. If we increase the intraspecific interaction coefficient, we can be sure that the growth of the prey population will both increase significantly. When the number of prey populations increases significantly, of course disease transmission and prey protection become determining factors, the continuation of the model in exosite interactions. In prey populations and susceptible prey to infection, growth does not require a long time compared to the growth of predator populations. The time required to achieve stable growth is rapid for the prey species. Although prey species' growth is more fluctuating compared to predator populations. Predatory species are more likely to be stable from the start of their growth. The significance of predatory growth is only at the beginning of growth, while after that it increases slowly and reaches an ideal equilibrium point. Each species has its own characteristics, so extensive studies are needed on more complex forms of response functions in further research. 
The Impact of Peer Pressure Mathematical Models Armed Criminal Groups with Criminal Mapping Area Pratama, Rian Ade; Ruslau, Maria F V
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 7, No 4 (2023): October
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v7i4.16255

Abstract

Model Armed Criminal Groups is mathematically realistic to be considered in the study of mathematical science. The aim of this research is to form a mathematical model of social cases of criminal acts. The given model is a criminal form that adopts the conformity of the conditions in the susceptible, exposed, infected, and recovered (SEIR) disease distribution model. The research method used is literature study and analysis. The research results show that there are 2 non-negative equilibrium, and one of them is stability analysis. Stability analysis is only carried out at equilibrium that does not contain a zero value with the Routh-Hurwitz criteria. In the results of other research the trajectories show that population growth tends not to experience fluctuations, this indicates that the population is growing towards stability rapidly. In case studies in the field, this marks a cycle of crime that quickly subsides or only occurs in a short period of time and does not occur in a sustainable manner. Overall the susceptible population, the exposed population, the infected population, and the recovered population experience the same conditions.
Analysis Dynamics Two Prey of a Predator-Prey Model with Crowley–Martin Response Function Pratama, Rian Ade; Toaha, Syamsuddin
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 7, No 3 (2023): July
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v7i3.14506

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.
Application of Beddington DeAngelis Response Function in Ecological Mathematical System: Study Fish Endemic Oliv Predator Species in Merauke Pratama, Rian Ade; Ruslau, Maria Fransina Veronica
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 6, No 1 (2022): January
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v6i1.5340

Abstract

Predator-prey type fishery models Oliv fish is a trans-endemic predator species that inhabits freshwater swamps and brackish water in Merauke, Papua. Maintaining the survival of the Oliv fish species is the main reason for compiling a mathematical model, so that it can be considered by local governments in making ecological policies. Method on model discussed is assembled with the growth of predator-prey populations following the growth of logistics. The response or predatory function corresponding to the behavior of endemic Oliv fish is the Beddington DeAngelis type. The growth of predatory species uses the concept of growth with stage structure, are divided into mature and immature. Research results show there are four equilibrium points of the mathematical model, but only one point becomes the asymptotic stable equilibrium point without harvesting W_4 (x^*,y^*,z^* )=92.823,1311.489,525.957 and equilibrium point with harvesting W_4 (x^*,y^*,z^* )=95.062,92.639,160.466 . Harvesting exploitation efforts are carried out by the community so that the harvesting variables are added with a proportional concept. Simulation of the results of the study shows a stable scheme and harvesting conducted can maintain the number of populations that continue. 
Co-Authors Amil Siddik, A. Muh. Aminah, R. Iin Siti Dadi, Oswaldus Henie Poerwandar Asmaningrum, Henie Poerwandar Ika Paridawati Iriani, Cindy Dian Kasbawati Kasbawati Maria Fransina Veronica Ruslau Meirista, Etriana Natsir, Irmawaty Nur?aini, Khumaeroh Dwi Nurbaiti Amir Nurhayati Nurhayati Nurhayati Palobo, Markus Ruslau, Maria F V Ruslau, Maria F V. Ruslau, Maria F. V. Ruslau, Maria Fransina Veronica Suryani, Dessy Rizki syamsuddin Toaha Taufik, Abdul Rachman Wahyudi, Candra Agus