Investigating the dynamics of Lotka$-$Volterra model with disease in the prey and predator species

Document Type : Research Paper


1 Esfarayen University of Technology‎, ‎Esfarayen‎, ‎North Khorasan‎, ‎Iran

2 Department of Mathematics‎, ‎University of Neyshabur‎, ‎Adib BLVD‎, ‎Neyshabur‎, ‎Iran‎


‎In this paper, a  predator$-$prey model  with logistic growth rate in the prey population was proposed.  It included an SIS infection in the prey and predator population.  The stability of the positive equilibrium point, the existence of Hopf and transcortical  bifurcation with parameter $a$ were investigated, where $a$ was regarded as  predation rate. It was found that when the parameter $a$ passed through a critical value,  stability changed and Hopf bifurcation occurred.  Biologically, the population  is positive and bounded. In the present article,  it was also shown that the model was bounded and that it had the positive solution.  Moreover, the current researchers came to the conclusion that although  the disease was present in the system, none of the species would be extinct. In other words, the system was persistent. Important thresholds, $R_{0}, R_{1}$ and $R_{2}$, were identified in the study. This theoretical study indicated that under certain conditions of $R_{0}, R_{1}$ and $R_{2}$,  the disease remained in the system or disappeared.


[1] R.M. Anderson and R.M. May, The invasion, persistence, and spread of infectious diseases within animal and plant communities, Phil. Trans. R. Sot. London B 314 (1986) 533–570.
[2] S. Beckerman, The equations of war, J. Curr. Anthropol. 32(5) (1991) 636–640.
[3] E. Bruce Nauman, Chemical Reactor Design, Optimization and Scaleup, Wiley, Hoboken, 2008.
[4] J. Cui, Y. Sun and H. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Diff. Equ. 20 (2008) 31–53.
[5] D.L. DeAngelis, S.M. Bartell and A.L. Brenkert, Effects of nutrient recycling and food-chain length on resilience, The Amer. Natur. 134(5) (1989) 778–805.
[6] H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Monogr. Textbooks Pure Appl. Math. 57, Marcel Dekker, New York, 1980.
[7] A. Ghasemabadi, Stability and bifurcation in a generalized delay prey–predator model, Nonlinear Dyn. 90 (2017) 2239–2251.
[8] J.M. Gottman, J.D. Murray, C.C. Swanson, R. Tyson and R.K. Swanson, The Mathematics of Marriage Dynamic Nonlinear Model, Cambridge Press, 2002.
[9] C. Grabner, H. Hahn, U. Leopold-Wildburger and S. Pickl, Analyzing the sustainability of harvesting behavior and the relationship to personality traits in a simulated Lotka–Volterra biotope, Eur. J. Oper. Res. 193 (2009) 761–767.
[10] K.P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol. 27 (1989) 609–631.
[11] C.S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomolog. Soc. Can. 45 (1965) 457–470.
[12] R. Memmarbashi, F. Alipour and A. Ghasemabadi, A nonstandard finite difference scheme for a SEI epidemic model, Punjab Univer. J. Math. 49(3) (2017) 133–147.
[13] B. Mukhopadhyay and R. Bhattacharyya, Dynamics of a delay-diffusion prey-predator model with disease in the prey, J. Appl. Math. Comput. 17(1-2) (2005) 361–377.
[14] L. Nunney, The stability of complex model ecosystems, Amer. Natur. 115(5) (1980) 639–649.
[15] L. Perko, Differential Equations and Dynamical System, Springer, New York, 2000.
[16] L. Randy, L. Haupt and S. E. Haupt, Practical genetic algorithms, Wiley, Hoboken, 2004.
[17] E. C. Pielou, Introduction to mathematical ecology, Wiley-Interscience, New York, 1982.
[18] P. Turchin, Complex Population Dynamics, A Theoretical/Empirical Syn-thesis. Princeton University Press, Princeton, NJ, 2003.
[19] E. Venturino, The influence of diseases on Lotka-Volterra systems, Rocky Mount. J. Math. 24 (1994) 381–402.
Volume 12, Issue 1
May 2021
Pages 633-648
  • Receive Date: 11 December 2017
  • Revise Date: 15 November 2018
  • Accept Date: 03 March 2019