Dynamics of a Fractional Order Eco-Epidemiological Model
DOI:
https://doi.org/10.11594/jtls.07.03.09Keywords:
Fractional order, eco-epidemiological model, Grünwald-Letnikov, behaviour solutionsAbstract
In this paper, we propose a fractional order eco-epidemiological model. We considere the existence of time memory in the growth rate of the three populations. We observed the dynamical behaviour by analysing with fractional order and then simulateing using Grünwald-Letnikov approximation to support analytical results. It found that the model has five equilibrium points, namely the origin, the survival of susceptible prey, the predator free equilibria, the infected prey free equilibria, the interior equilibria. Numerical simulations show that the existence of fractional order  is a factor which affects the behaviour of solutions.
Â
References
Brauer F, Castillo-Chavez C (2011) Mathematical models in population biology and epidemiology. 2nd Edition. New York, Springer.
Rosenzweig ML, MacArthur RH (1963) Graphical representation and stability condition of predator-prey interactions. American Naturalist 97 (895): 209 – 223. doi: 10.1086/282272.
Hadeler KP, Freedman HI (1989) Predator-prey populations with parasitic infection. Journal of Mathematical Biology 27 (6): 609 – 631. doi: 10.1007/BF00276947.
Suryanto A (2017) Dynamics of an eco-epidemiological model with saturated incidence rate, AIP Conference Proceedings 1825: 020021. doi: 10.1063/1.4978990.
Chattopadhyay J, Arino O (1999) A Predator-prey model with disease in the prey. Nonlinear Analysis: Theory, Methods and Applications 36 (6): 747 – 766. doi: 10.1016/S0362-546X(98)00126-6.
Saifuddin Md, Sasmal SK, Biswas S et al. (2015) Effect of emergent carrying capacity in an eco-epidemiological sys-tem. Mathematical Methods in the Applied Sciences 39 (4): 806 – 823. doi: 10.1002/mma.3523.
Saifuddin Md, Biswas S, Samanta S et al. (2016) Complex dynamics of an eco-epidemiological model with different competition coefficients and weak alle in the predator. Chaos, Solitons and Fractals 91: 270 – 285. doi: 10.1016/j.chaos.2016.06.009.
Diethelm K (2010) The analysis of fractional differential equations. Berlin, Springer-Verlag.
Petras I (2011) Fractional-order nonlinear systems: Modeling, analysis and simulation Berlin, Springer-Verlag. doi: 10.1007/978-3-642-18101-6.
Rihan FA, Lakshmanan S, Hashish AH et al. (2015) Fractional-order delayed predator-prey systems with Holling type-II functional response. Nonlinear Dynamics 80 (1 – 2): 777 – 789. doi: 10.1007/s11071-015-1905-8.
Suryanto A, Darti I, Anam S (2017) Stability analysis of a fractional order modified Leslie-Gower model additive Allee effects. International Journal of Mathematics and Mathematical Sciences 2017 (2017). doi: 10.1155/2017/8273430.
Rivero M, Trujillo JJ, Vázquez L, Velasco MP (2011) Fractional dynamics of populations. Applied Mathematics and Computation 218 (3): 1089 – 1095. doi: 10.1016/j.amc.2011.03.017.
Ghaziani RK, Alidousti J, Eshkaftaki AB (2016) Stability and dynamics of fractional order Leslie-Gower prey predator model. Applied Mathematical Modelling 40 (3): 2075 – 2086. doi: 10.1016/j.apm.2015.09.014.
Rida S, Khalil ZM, Hosham HA, Gadellah S, (2014) Predator-prey fractional-order dynamical system with both the population affected by diseases. Journal of Fractional Cal-culus and Applications 5 (13): 1 – 11.
Downloads
Published
Issue
Section
License
The work has not been published before (except in the form of an abstract or part of a published lecture or thesis) and it is not under consideration for publication elsewhere. When the manuscript is accepted for publication in this journal, the authors agree to automatic transfer of the copyright to the publisher.
Journal of Tropical Life Science is licensed under Creative Commons Attribution-NonCommercial 4.0 International License