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Dynamics of a Fractional-Order Predator-Prey Model with Infectious Diseases in Prey Panigoro, Hasan S.; Suryanto, Agus; Kusumahwinahyu, Wuryansari Muharini; Darti, Isnani
Communication in Biomathematical Sciences Vol 2, No 2 (2019)
Publisher : Indonesian Bio-Mathematical Society

In this paper, a dynamical analysis of a fractional-order predator-prey model with infectious diseases in prey is performed. First, we prove the existence, uniqueness, non-negativity, and boundedness of the solution. We also show that the model has at most five equilibrium points, namely the origin, the infected prey and predator extinction point, the infected prey extinction point, the predator extinction point, and the co-existence point. For the first four equilibrium points, we show that the local stability properties of the fractional-order system are the same as the first-order system, but for the co-existence point, we have different local stability properties.We also present the global stability of each equilibrium points except for the origin point. We observe an interesting phenomenon, namely the occurrence of Hopf bifurcation around the co-existence equilibrium point driven by the order of fractional derivative. Moreover, we show some numerical simulations based on a predictor-corrector scheme to illustrate the result of our dynamical analysis.

The Influence of Additive Allee Effect and Periodic Harvesting to the Dynamics of Leslie-Gower Predator-Prey Model Hasan S. Panigoro; Emli Rahmi; Novianita Achmad; Sri Lestari Mahmud
Jambura Journal of Mathematics Vol 2, No 2: Juli 2020
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

In this paper, the influence of additive Allee effect in prey and periodic harvesting in predator to the dynamics of the Leslie-Gower predator-prey model is proposed. We first simplify the model to the non-dimensional system by scaling the variable and transform the model into an autonomous system. If the effect Allee is weak, we have at most two equilibrium points, else if the Allee effect is strong, at most four equilibrium points may exist. Furthermore, the behavior of the system around equilibrium points is investigated. In the end, we give numerical simulations to support theoretical results.

Complexity of a Discrete-Time Predator-Prey Model Involving Prey Refuge Proportional to Predator P. K. Santra; Hasan S. Panigoro; G. S. Mahapatra
Jambura Journal of Mathematics Vol 4, No 1: January 2022
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

In this paper, a discrete-time predator-prey model involving prey refuge proportional to predator density is studied. It is assumed that the rate at which prey moves to the refuge is proportional to the predator density. The fixed points, their local stability, and the existence of Neimark-Sacker bifurcation are investigated. At last, the numerical simulations consisting of bifurcation diagrams, phase portraits, and time-series are given to support analytical findings. The occurrence of chaotic solutions are also presented by showing the Lyapunov exponent while some parameters are varied.

The Dynamics of a Discrete Fractional-Order Logistic Growth Model with Infectious Disease Hasan S Panigoro; Emli Rahmi
Contemporary Mathematics and Applications (ConMathA) Vol. 3 No. 1 (2021)
Publisher : Universitas Airlangga

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20473/conmatha.v3i1.26938

In this paper, we study the dynamics of a discrete fractional-order logistic growth model with infectious disease. We obtain the discrete model by applying the piecewise constant arguments to the fractional-order model. This model contains three fixed points namely the origin point, the disease-free point, and the endemic point. We confirm that the origin point is always exists and unstable, the disease-free point is always exists and conditionally stable, and the endemic point is conditionally exists and stable. We also investigate the existence of forward, period-doubling, and Neimark-Sacker bifurcation. The numerical simulations are also presented to confirm the analytical results. We also show numerically the existence of period-3 solution which leads to the occurrence of chaotic behavior.

Dynamics of a Fractional-Order Predator-Prey Model with Infectious Diseases in Prey Hasan S. Panigoro; Agus Suryanto; Wuryansari Muharini Kusumahwinahyu; Isnani Darti
Communication in Biomathematical Sciences Vol. 2 No. 2 (2019)
Publisher : Indonesian Bio-Mathematical Society

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/cbms.2019.2.2.4

In this paper, a dynamical analysis of a fractional-order predator-prey model with infectious diseases in prey is performed. First, we prove the existence, uniqueness, non-negativity, and boundedness of the solution. We also show that the model has at most five equilibrium points, namely the origin, the infected prey and predator extinction point, the infected prey extinction point, the predator extinction point, and the co-existence point. For the first four equilibrium points, we show that the local stability properties of the fractional-order system are the same as the first-order system, but for the co-existence point, we have different local stability properties.We also present the global stability of each equilibrium points except for the origin point. We observe an interesting phenomenon, namely the occurrence of Hopf bifurcation around the co-existence equilibrium point driven by the order of fractional derivative. Moreover, we show some numerical simulations based on a predictor-corrector scheme to illustrate the result of our dynamical analysis.

Forecasting COVID-19 Epidemic in Spain and Italy Using A Generalized Richards Model with Quantified Uncertainty Isnani Darti; Agus Suryanto; Hasan S. Panigoro; Hadi Susanto
Communication in Biomathematical Sciences Vol. 3 No. 2 (2020)
Publisher : Indonesian Bio-Mathematical Society

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/cbms.2020.3.2.1

The Richards model and its generalized version are deterministic models that are often implemented to fit and forecast the cumulative number of infective cases in an epidemic outbreak. In this paper we employ a generalized Richards model to predict the cumulative number of COVID-19 cases in Spain and Italy, based on available epidemiological data. To quantify uncertainty in the parameter estimation, we use a parametric bootstrapping approach to construct a 95% confidence interval estimation for the parameter model. Here we assume that the time series data follow a Poisson distribution. It is found that the 95% confidence interval of each parameter becomes narrow with the increasing number of data. All in all, the model predicts daily new cases of COVID-19 reasonably well during calibration periods. However, the model fails to produce good forecasts when the amount of data used for parameter estimations is not sufficient. Based on our parameter estimates, it is found that the early stages of COVID-19 epidemic, both in Spain and in Italy, followed an almost exponentially growth. The epidemic peak in Spain and Italy is respectively on 2 April 2020 and 28 March 2020. The final sizes of cumulative number of COVID-19 cases in Spain and Italy are forecasted to be at 293220 and 237010, respectively.

Bifurkasi Hopf pada Model Lotka-Volterra Orde-Fraksional dengan Efek Allee Aditif pada Predator Hasan S. Panigoro; Dian Savitri
Jambura Journal of Biomathematics (JJBM) Volume 1, Issue 1: June 2020
Publisher : Department of Mathematics, State University of Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.34312/jjbm.v1i1.6908

This article aims to study the dynamics of a Lotka-Volterra predator-prey model with Allee effect in predator. According to the biological condition, the Caputo fractional-order derivative is chosen as its operator. The analysis is started by identifying the existence, uniqueness, and non-negativity of the solution. Furthermore, the existence of equilibrium points and their stability is investigated. It has shown that the model has two equilibrium points namely both populations extinction point which is always a saddle point, and a conditionally stable co-existence point, both locally and globally. One of the interesting phenomena is the occurrence of Hopf bifurcation driven by the order of derivative. Finally, the numerical simulations are given to validate previous theoretical results.

Bifurkasi Hopf pada model prey-predator-super predator dengan fungsi respon yang berbeda Dian Savitri; Hasan S. Panigoro
Jambura Journal of Biomathematics (JJBM) Volume 1, Issue 2: December 2020
Publisher : Department of Mathematics, State University of Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.34312/jjbm.v1i2.8399

This article discusses the one-prey, one-predator, and the super predator model with different types of functional response. The rate of prey consumption by the predator follows Holling type I functional response and the rate of predator consumption by the super predator follows Holling type II functional response. We identify the existence and stability of critical points and obtain that the extinction of all population points is always unstable, and the other two are conditionally stable i.e., the super predator extinction point and the co-existence point. Furthermore, we give the numerical simulations to describe the bifurcation diagram and phase portraits of the model. The bifurcation diagram is obtained by varying the parameter of the conversion rate of predator biomass into a new super-predator which gives forward and Hopf bifurcation. The forward bifurcation occurs around the super predator extinction point while Hopf bifurcation occurs around the interior of the model. Based on the terms of existence and numerical simulation, we confirm that the conversion rate of predator biomass into a new super-predator controls the dynamics of the system and maintains the existence of predator.

Computational dynamics of a Lotka-Volterra Model with additive Allee effect based on Atangana-Baleanu fractional derivative Hasan S. Panigoro; Emli Rahmi
Jambura Journal of Biomathematics (JJBM) Volume 2, Issue 2: December 2021
Publisher : Department of Mathematics, State University of Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.34312/jjbm.v2i2.11886

This paper studies an interaction between one prey and one predator following Lotka-Volterra model with additive Allee effect in predator. The Atangana-Baleanu fractional-order derivative is used for the operator. Since the theoretical ways to investigate the model using this operator are limited, the dynamical behaviors are identified numerically. By simulations, the influence of the order of the derivative on the dynamical behaviors is given. The numerical results show that the order of the derivative may impact the convergence rate, the occurrence of Hopf bifurcation, and the evolution of the diameter of the limit-cycle.

Revitalisasi Danau Limboto dengan Pengerukan Endapan di Danau: Pemodelan, Analisis, dan Simulasinya Sri Lestari Mahmud; Novianita Achmad; Hasan S. Panigoro
Jambura Journal of Biomathematics (JJBM) Volume 1, Issue 1: June 2020
Publisher : Department of Mathematics, State University of Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.34312/jjbm.v1i1.6945

Limboto lake is one of assets of Province of Gorontalo that provides many benefits to the surrounding society. The main problem of Limboto lake is the silting of the lake due to sedimentation caused by forest erosion, household waste, water hyacinth, and fish farming which is not environmentally friendly. In this article, a mathematical approach is used to modeling the Limboto lake siltation by including the revitalization solution namely the lake dredging. Mathematical modeling begins by building and limiting assumptions, constructing variables and parameters in mathematical symbols, and forming them into a first order differential equation system deterministically. Furthermore, we study the dynamics of the model such as identifying the existence of equilibrium points and their stability conditions. We also give a numerical simulations to show the conditions based on the stability requirements in previous analytical results.

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