Stability analysis and nonstandard Grünwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response

Agus Suryanto; Isnani Darti

doi:10.1063/1.5016645

Stability analysis and nonstandard Grünwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response

Agus Suryanto^*
, Isnani Darti

^*Corresponding author for this work

Faculty of Mathematics and Natural Sciences

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

9 Citations (Scopus)

Abstract

In this paper we discuss a fractional order predator-prey model with ratio-dependent functional response. The dynamical properties of this model is analyzed. Here we determine all equilibrium points of this model including their existence conditions and their stability properties. It is found that the model has two type of equilibria, namely the predator-free point and the co-existence point. If there is no co-existence equilibrium, i.e. when the coefficient of conversion from the functional response into the growth rate of predator is less than the death rate of predator, then the predator-free point is asymptotically stable. On the other hand, if the co-existence point exists then this equilibrium is conditionally stable. We also construct a nonstandard Grnwald-Letnikov (NSGL) numerical scheme for the propose model. This scheme is a combination of the Grnwald-Letnikov approximation and the nonstandard finite difference scheme. This scheme is implemented in MATLAB and used to perform some simulations. It is shown that our numerical solutions are consistent with the dynamical properties of our fractional predator-prey model.

Original language	English
Title of host publication	International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017
Editors	Adem Kilicman, Marjono, Ratno Bagus Edy Wibowo, Moch. Aruman Imron
Publisher	American Institute of Physics Inc.
ISBN (Electronic)	9780735416055
DOIs	https://doi.org/10.1063/1.5016645
Publication status	Published - 5 Dec 2017
Event	International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017 - Malang, Indonesia Duration: 2 Aug 2017 → 3 Aug 2017

Publication series

Name	AIP Conference Proceedings
Volume	1913
ISSN (Print)	0094-243X
ISSN (Electronic)	1551-7616

Conference

Conference	International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017
Country/Territory	Indonesia
City	Malang
Period	2/08/17 → 3/08/17

UN SDGs

This output contributes to the following UN Sustainable Development Goals (SDGs)

SDG 15 Life on Land

Keywords

Fractional order predator-prey model
local stability
nonstandard Grnwald-Letnikov scheme

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10.1063/1.5016645

Cite this

Suryanto, A., & Darti, I. (2017). Stability analysis and nonstandard Grünwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response. In A. Kilicman, Marjono, R. B. E. Wibowo, & M. A. Imron (Eds.), International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017 Article 020011 (AIP Conference Proceedings; Vol. 1913). American Institute of Physics Inc.. https://doi.org/10.1063/1.5016645

Suryanto, Agus ; Darti, Isnani. / Stability analysis and nonstandard Grünwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response. International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017. editor / Adem Kilicman ; Marjono ; Ratno Bagus Edy Wibowo ; Moch. Aruman Imron. American Institute of Physics Inc., 2017. (AIP Conference Proceedings).

@inproceedings{c83e08ada81c4032b4ea2c22e4b9e4f6,

title = "Stability analysis and nonstandard Gr{\"u}nwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response",

abstract = "In this paper we discuss a fractional order predator-prey model with ratio-dependent functional response. The dynamical properties of this model is analyzed. Here we determine all equilibrium points of this model including their existence conditions and their stability properties. It is found that the model has two type of equilibria, namely the predator-free point and the co-existence point. If there is no co-existence equilibrium, i.e. when the coefficient of conversion from the functional response into the growth rate of predator is less than the death rate of predator, then the predator-free point is asymptotically stable. On the other hand, if the co-existence point exists then this equilibrium is conditionally stable. We also construct a nonstandard Grnwald-Letnikov (NSGL) numerical scheme for the propose model. This scheme is a combination of the Grnwald-Letnikov approximation and the nonstandard finite difference scheme. This scheme is implemented in MATLAB and used to perform some simulations. It is shown that our numerical solutions are consistent with the dynamical properties of our fractional predator-prey model.",

keywords = "Fractional order predator-prey model, local stability, nonstandard Grnwald-Letnikov scheme",

author = "Agus Suryanto and Isnani Darti",

note = "Publisher Copyright: {\textcopyright} 2017 Author(s).; International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017 ; Conference date: 02-08-2017 Through 03-08-2017",

year = "2017",

month = dec,

day = "5",

doi = "10.1063/1.5016645",

language = "English",

series = "AIP Conference Proceedings",

publisher = "American Institute of Physics Inc.",

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Suryanto, A & Darti, I 2017, Stability analysis and nonstandard Grünwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response. in A Kilicman, Marjono, RBE Wibowo & MA Imron (eds), International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017., 020011, AIP Conference Proceedings, vol. 1913, American Institute of Physics Inc., International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017, Malang, Indonesia, 2/08/17. https://doi.org/10.1063/1.5016645

Stability analysis and nonstandard Grünwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response. / Suryanto, Agus ; Darti, Isnani.
International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017. ed. / Adem Kilicman; Marjono; Ratno Bagus Edy Wibowo; Moch. Aruman Imron. American Institute of Physics Inc., 2017. 020011 (AIP Conference Proceedings; Vol. 1913).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Stability analysis and nonstandard Grünwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response

AU - Suryanto, Agus

AU - Darti, Isnani

PY - 2017/12/5

Y1 - 2017/12/5

N2 - In this paper we discuss a fractional order predator-prey model with ratio-dependent functional response. The dynamical properties of this model is analyzed. Here we determine all equilibrium points of this model including their existence conditions and their stability properties. It is found that the model has two type of equilibria, namely the predator-free point and the co-existence point. If there is no co-existence equilibrium, i.e. when the coefficient of conversion from the functional response into the growth rate of predator is less than the death rate of predator, then the predator-free point is asymptotically stable. On the other hand, if the co-existence point exists then this equilibrium is conditionally stable. We also construct a nonstandard Grnwald-Letnikov (NSGL) numerical scheme for the propose model. This scheme is a combination of the Grnwald-Letnikov approximation and the nonstandard finite difference scheme. This scheme is implemented in MATLAB and used to perform some simulations. It is shown that our numerical solutions are consistent with the dynamical properties of our fractional predator-prey model.

AB - In this paper we discuss a fractional order predator-prey model with ratio-dependent functional response. The dynamical properties of this model is analyzed. Here we determine all equilibrium points of this model including their existence conditions and their stability properties. It is found that the model has two type of equilibria, namely the predator-free point and the co-existence point. If there is no co-existence equilibrium, i.e. when the coefficient of conversion from the functional response into the growth rate of predator is less than the death rate of predator, then the predator-free point is asymptotically stable. On the other hand, if the co-existence point exists then this equilibrium is conditionally stable. We also construct a nonstandard Grnwald-Letnikov (NSGL) numerical scheme for the propose model. This scheme is a combination of the Grnwald-Letnikov approximation and the nonstandard finite difference scheme. This scheme is implemented in MATLAB and used to perform some simulations. It is shown that our numerical solutions are consistent with the dynamical properties of our fractional predator-prey model.

KW - Fractional order predator-prey model

KW - local stability

KW - nonstandard Grnwald-Letnikov scheme

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U2 - 10.1063/1.5016645

DO - 10.1063/1.5016645

M3 - Conference contribution

AN - SCOPUS:85037819356

T3 - AIP Conference Proceedings

BT - International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017

A2 - Kilicman, Adem

A2 - Marjono, null

A2 - Wibowo, Ratno Bagus Edy

A2 - Imron, Moch. Aruman

PB - American Institute of Physics Inc.

T2 - International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017

Y2 - 2 August 2017 through 3 August 2017

ER -

Suryanto A , Darti I. Stability analysis and nonstandard Grünwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response. In Kilicman A, Marjono, Wibowo RBE, Imron MA, editors, International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017. American Institute of Physics Inc. 2017. 020011. (AIP Conference Proceedings). doi: 10.1063/1.5016645

Stability analysis and nonstandard Grünwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response

Abstract

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Keywords

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