# The Existence of a Limit-Cycle of a Discrete-Time Lotka-Volterra Model with Fear Effect and Linear Harvesting

**Authors**: Hasan S. Panigoro, Resmawan Resmawan, Emli Rahmi, Muhammad Afrizal Beta, Amelia Tri Rahma Sidik

Modeling the interaction between prey and predator plays an important role in maintaining the balance of the ecological system. In this paper, a discrete-time mathematical model is constructed via a forward Euler scheme, and then studied the dynamics of the model analytically and numerically. The analytical results show that the model has two fixed points, namely the origin and the interior points. The possible dynamical behaviors are shown analytically and demonstrated numerically using some phase portraits. We show numerically that the model has limit-cycles on its interior. This guarantees that there exists a condition where both prey and predator maintain their existence periodically.

E3S Web of Conferences 400, 03003 (2023)

# DYNAMIC ANALYSIS OF THE MATHEMATICAL MODEL FOR THE CHOLERA DISEASE SPREAD INVOLVING MEDICATION AND ENVIRONMENTAL SANITATION

**Authors**: R Resmawan, Lailany Yahya, Sri Lestari Mahmud, Agusyarif Rezka Nuha, Nazrilla Hasan Laita

This study aims to analyze the mathematical model of the cholera disease spread involving medicationnd environmental sanitation. The model was analyzed by determining the equilibrium point and the basic reproduction number. The next step was to analyze the equilibrium point, sensitivity, and simulate numerically. Analysis of the stability of the disease-free and endemic equilibrium points usedhe Routh-Hurwitz criteria and the Castillo-Chaves and Song Theorem. The Analysis resultf the model produced two equilibrium points; namely the disease-freequilibrium point for local asymptotic stability and the endemic equilibrium point for local asymptotic stability if . Furthermore, the sensitivity analysis indicated the most sensitive parameters for basic reproductive number changes in succession are the parameters for natural birth rates , the transmission rate of bacteria from the environment to humans , the saturated concentration of bacteria in water , an increase in the bacterial population caused by environmental pollution rate by humans . Numerical simulations suggest an increase to give vaccine can contribute to slowing the transmission of cholera where as the reduction of a vaccine able to promote the transmission of cholera diseases.

VOL 17 NO 1 (2023): BAREKENG: JOURNAL OF MATHEMATICS AND ITS APPLICATIONS

# Analisis Dinamik pada Model Matematika SVEIBR dengan Kontrol Optimal Untuk Pengendalian Penyebaran Penyakit Kolera

**Authors**: Agusyarif Rezka Nuha, Resmawan Resmawan, Sri Lestari Mahmud, Asriadi Asriadi, Andi Agung, Sri Istiyarti Uswatun Chasanah

Cholera is an infectious disease that attacks the human digestive system and can cause death. This article discusses the research results related to the mathematical model of the spread of cholera in the form of an optimal control system by combining three control strategies: vaccination, quarantine, and environmental sanitation. Pontryagin's maximum principle is applied to obtain optimal conditions based on the control strategy applied. Referring to the optimal conditions set, the model was solved numerically using the Runge-Kutta Order 4 method to describe the theoretical results. The calculation results show that applying the three control strategies in controlling the spread of cholera positively impacts reducing the number of cases of infection so that disease transmission can be discontinued.

# The existence of Neimark-Sacker bifurcation on a discrete-time SIS-Epidemic model incorporating logistic growth and allee effect

**Authors**: Amelia Tri Rahma Sidik, Hasan S. Panigoro, Resmawan Resmawan, Emli Rahmi

In this article, the dynamical properties of a discrete-time SIS-Epidemic model incorporating logistic growth rate and Allee effect are investigated. The forward Euler discretization method is employed to obtain the discrete-time model. All possible fixed points are identified, including their local dynamics and the existence of two important phenomena, namely transcritical and period-doubling bifurcation. Some numerical simulations are explored to show the analytical findings, such as bifurcation diagrams, phase portraits, and time series. The occurrence of a period-3 window is shown numerically, which routes to chaotic solutions.

# Bifurcation Analysis of a Predator-Prey Model Involving Age Structure, Intraspecific Competition, Michaelis-Menten Type Harvesting, and Memory Effect

**Authors:** *HS Panigoro, E Rahmi, R Resmawan*

*The complexity of the dynamical behaviors of interaction between prey and its predator is studied. The prey and predator relationship involves the age structure and intraspecific competition on predators, and the nonlinear harvesting of prey following the Michaelis-Menten type term. Some biological validities are shown for the constructed model such as the existence and uniqueness as well as the non-negativity and boundedness of solutions. Three equilibrium points namely the origin, axial, and interior point are found including their global dynamics by employing the Lyapunov function along with the generalized Lassale invariant principle. The changes in dynamical behaviors driven by the harvesting and the memory effect are exhibited namely transcritical, saddle-node, backward, and Hopf bifurcations. The appearance of these interesting phenomena is strengthened by giving numerical simulations consisting of bifurcation diagrams, phase portraits, and their time series.*

Frontiers in Applied Mathematics and Statistics, 124, 2022

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