Intra-Specific Competition in Prey Can Control Chaos in a Prey-Predator Model

  • Md Saifuddin
  • Santanu Biswas
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)


In this article, we propose a general prey-predator model with the presence of Allee effect in prey. We have considered different competition coefficients within the prey population, which leads to the emergent carrying capacity. The stability analysis of the system is discussed. Further, the dynamical behavior of the system is analyzed, taking delay and emergent carrying capacity as bifurcation parameters. Time delay can turn a stable equilibrium into an unstable one. It was shown that our system experiences the Hopf bifurcation, as the delay parameter crosses some critical values. Further increases in delay produce chaos, which can be controlled by the emergent carrying capacity.


The Allee effect Delay Stability analysis Eco-epidemiological system 


  1. Allee, W. C. (1931). Animal aggregations. A study in general sociology. Chicago: University of Chicago Press.CrossRefGoogle Scholar
  2. Biswas, S., Sasmal, K. S., Samanta, S., Saifuddin, M., Khan, Q. J. A., Alquranc, M., & Chattopadhyaya, J. (2015). A delayed eco-epidemiological system with infected prey and predator subject to the weak Allee effect. Mathematical Biosciences, 263, 198–208.MathSciNetCrossRefGoogle Scholar
  3. Biswas, S., Saifuddin, M., Sasmal, K. S., Samanta, S., Pal, N., Ababneh, F., & Chattopadhyaya, J. (2016). A delayed prey-predator system with prey subject to the strong Allee effect and disease. Nonlinear Dynamics, 84, 1569–1594.MathSciNetCrossRefGoogle Scholar
  4. Celik, C., Merdan, H., Duman, O., & Akin, O. (2008). Allee effects on population dynamics with delay. Chaos, Solitons & Fractals, 37, 65–74.ADSMathSciNetCrossRefGoogle Scholar
  5. Gopalsamy, K. (1992). Stability and oscillation in delay differential equation of population dynamics. Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
  6. Hilker, F. M., Langlais, M., Petrovskii, S. V., & Malchow, H. (2007). A diffusive SI model with Allee effect and application to FIV. Mathematical Biosciences, 206, 61–80.MathSciNetCrossRefGoogle Scholar
  7. Kuang, Y. (1993). Delay differential equation with applications in population dynamics. New York: Academic.zbMATHGoogle Scholar
  8. MacDonald, N. (1989). Biological delay systems: Linear stability theory. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  9. Pal, P. J., Saha, T., Sen, M., & Banerjee, M. (2012). A delayed predator–prey model with strong Allee effect in prey population growth. Nonlinear Dynamics, 68, 23–42.MathSciNetCrossRefGoogle Scholar
  10. Yan, J., Zhao, A., & Yan, W. (2005) Existence and global attractivity of periodic solution for an impulsive delay differential equation with Allee effect. Journal of Mathematical Analysis and Applications, 309, 489–504.ADSMathSciNetCrossRefGoogle Scholar
  11. Zhang, T., Zang, H. (2014). Delay-induced Turing instability in reaction-diffusion equations. Physical Review E, 90, 052908.ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Md Saifuddin
    • 1
  • Santanu Biswas
    • 2
  1. 1.Department of MathematicsBidhan Chandra CollegeHooghlyIndia
  2. 2.Department of MathematicsAdamas UniversityBarasatIndia

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