### Dynamics of a Fractional-Order Predator-Prey Model with Infectious Diseases in Prey

Hasan S. Panigoro, Agus Suryanto, Wuryansari Muharini Kusumahwinahyu, Isnani Darti

#### Abstract

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.

#### Keywords

fractional-order; hopf bifurcation; infectious diseases; predator-prey; stability

PDF

#### References

Ahmed, E., El-Sayed, A.M.A. and El-Saka, H.A., 2006. On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, R¨ossler, Chua and Chen systems. Physics Letters A, 358(1), pp.1-4.

Almeida, R., Bastos, N.R. and Monteiro, M.T.T., 2016. Modeling some real phenomena by fractional differential equations. Mathematical Methods in the Applied Sciences, 39(16), pp.4846-4855.

Alzahrani, A.K., Alshomrani, A.S., Pal, N. and Samanta, S., 2018. Study of an eco-epidemiological model with Z-type control. Chaos, Solitons & Fractals, 113, pp.197-208.

Choi, S.K., Kang, B. and Koo, N., 2014. Stability for Caputo fractional differential systems. In Abstract and Applied Analysis (Vol. 2014). Hindawi.

Das, M., Maiti, A. and Samanta, G.P., 2018. Stability analysis of a prey-predator fractional order model incorporating prey refuge. Ecological Genetics and Genomics, 7, pp.33-46.

Diethelm, K., Ford, N.J. and Freed, A.D., 2002. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1-4), pp.3-22.

Elsadany, A.A. and Matouk, A.E., 2015. Dynamical behaviors of fractional-order Lotka–Volterra predator–prey model and its discretization. Journal of Applied Mathematics and Computing, 49(1-2), pp.269-283.

Huda, M.N., Trisilowati, T. and Suryanto, A., 2017. Dynamical analysis of fractional-order Hastings-Powell food chain model with alternative food. The Journal of Experimental Life Science, 7(1), pp.39-44.

Huo, J., Zhao, H. and Zhu, L., 2015. The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Analysis: Real World Applications, 26, pp.289-305.

Ghaziani, R.K., Alidousti, J. and Eshkaftaki, A.B., 2016. Stability and dynamics of a fractional order Leslie–Gower prey–predator model. Applied Mathematical Modelling, 40(3), pp.2075-2086.

Li, H.L., Zhang, L., Hu, C., Jiang, Y.L. and Teng, Z., 2017. Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing, 54(1-2), pp.435-449.

Li, X. and Wu, R., 2014. Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system. Nonlinear Dynamics, 78(1), pp.279-288.

Li, Y., Chen, Y. and Podlubny, I., 2010. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Computers & Mathematics with Applications, 59(5), pp.1810-1821.

Liu, X., Hong, L., Yang, L. and Tang, D., 2019. Bifurcations of a New Fractional-Order System with a One-Scroll Chaotic Attractor. Discrete Dynamics in Nature and Society, 2019.

Matignon, D., 1996, July. Stability results for fractional differential equations with applications to control processing. In Computational engineering in systems applications (Vol. 2, pp. 963-968).

Mondal, S., Lahiri, A. and Bairagi, N., 2017. Analysis of a fractional order eco-epidemiological model with prey infection and type 2 functional response. Mathematical Methods in the Applied Sciences, 40(18), pp.6776-6789.

Nosrati, K. and Shafiee, M., 2018. Fractional-order singular logistic map: Stability, bifurcation and chaos analysis. Chaos, Solitons & Fractals, 115, pp.224-238.

Nugraheni, K., Trisilowati, T. and Suryanto, A., 2017. Dynamics of a Fractional Order Eco-Epidemiological Model. Journal of Tropical Life Science, 7(3), pp.243-250.

Petr´aˇs, I., 2011. Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media.

Purnomo, A.S., Darti, I. and Suryanto, A., 2017, December. Dynamics of eco-epidemiological model with harvesting. In AIP Conference Proceedings (Vol. 1913, No. 1, p. 020018). AIP Publishing.

Rida, S., Khalil, Z.M., Hosham, H.A. and 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), pp.1-11.

Saifuddin, M., Biswas, S., Samanta, S., Sarkar, S. and Chattopadhyay, J., 2016. Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator. Chaos, Solitons & Fractals, 91, pp.270-285.

Satriyantara, R., Suryanto, A. and Hidayat, N., 2018. Numerical Solution of a Fractional-Order Predator-Prey Model with Prey Refuge and Additional Food for Predator. The Journal of Experimental Life Science, 8(1), pp.66-70.

Suryanto, A., 2017, March. Dynamics of an eco-epidemiological model with saturated incidence rate. In AIP Conference Proceedings (Vol. 1825, No. 1, p. 020021). AIP Publishing.

Suryanto, A. and Darti, I., 2017, December. Stability analysis and nonstandard Gr¨unwald-Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response. In AIP Conference Proceedings (Vol. 1913, No. 1, p. 020011). AIP Publishing.

Suryanto, A. and Darti, I., 2019. Dynamics of Leslie-Gower Pest-Predator Model with Disease in Pest Including Pest-Harvesting and Optimal Implementation of Pesticide. International Journal of Mathematics and Mathematical Sciences, 2019.

Suryanto, A., Darti, I. and Anam, S., 2017. Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect. International Journal of Mathematics and Mathematical Sciences, 2017.

Vargas-De-Le´on, C., 2015. Volterra-type Lyapunov functions for fractional-order epidemic systems. Communications in Nonlinear Science and Numerical Simulation, 24(1-3), pp.75-85.

Wei, Z., Li, Q. and Che, J., 2010. Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative. Journal of Mathematical Analysis and Applications, 367(1), pp.260-272.

Wituła, R. and Słota, D., 2010. Cardano’s formula, square roots, Chebyshev polynomials and radicals. Journal of Mathematical Analysis and Applications, 363(2), pp.639-647.

Xie, Y., Lu, J. and Wang, Z., 2019. Stability analysis of a fractional-order diffused prey–predator model with prey refuges. Physica A: Statistical Mechanics and its Applications, 526, p.120773.

DOI: http://dx.doi.org/10.5614%2Fcbms.2019.2.2.4

### Refbacks

• There are currently no refbacks. 