Dynamical behavior of predator–prey model with non-smooth prey harvesting

The objective of the current paper is to investigate the dynamics of a new predator–prey model, where the prey species obeys the law of logistic growth and is subjected to a non-smooth switched harvest: when the density of the prey is below a switched value, the harvest has a linear rate.  Otherwise, the harvesting rate is constant.  The equilibria of the proposed system are described, and the boundedness of its solutions is examined.  We discuss the existence of periodic solutions; we show the appearance of two limit cycles, an unstable inner limit cycle and a stable outer one.  As the values of the model parameters vary, several kinds of bifurcation for the model are detected, such as transcritical, saddle–node, and Hopf bifurcations.  Finally, some numerical examples of the model are performed to confirm the theoretical results obtained.

predator–prey model
switched harvest
стабільність
біфуркація
limit cycle
  1. Berryman A. A.   The origins and evolution of predator–prey theory.  Ecology.  73 (5), 1530–1535 (1992).
  2. Hafdane M., Agmour I., El Foutayeni Y.  Study of Hopf bifurcation of delayed tritrophic system: dinoflagellates, mussels, and crabs.  Mathematical Modeling and Computing.  10 (1), 66–79 (2023).
  3. Kar T. K.  Modelling and analysis of a harvested prey–predator system incorporating a prey refuge.  Journal of Computational and Applied Mathematics.  185 (1), 19–33 (2006).
  4. Leard B., Lewis C., Rebaza J.  Dynamics of ratio-dependent Predator–Prey models with nonconstant harvesting.  Discrete and Continuous Dynamical Systems – S.  1 (2), 303–315 (2008).
  5. Lenzini P., Rebaza J.  Nonconstant predator harvesting on ratio-dependent predator–prey models.  Applied Mathematical Sciences.  4 (16), 791–803 (2010).
  6. Li B., Liu S., Cui J., Li J.  A simple predator-prey population with rich dynamics.  Applied Sciences.  6 (5), 151 (2016).
  7. Liu X., Huang Q.  Comparison and analysis of two forms of harvesting functions in the two-prey and one-predator model.  Journal of Inequalities and Applications.  2019, 307 (2019).
  8. Lv Y., Yuan R., Pei Y.  Two types of predator–prey models with harvesting: Non-smooth and non-continuous.  Journal of Computational and Applied Mathematics.  250, 122–142 (2013).
  9. Xiao M., Cao J.  Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator-prey model with linear harvesting rate: Analysis and computation.  Mathematical and Computer Modelling.  50 (3–4), 360–379 (2009).
  10. Xiao D., Jennings L. S.  Bifurcations of a ratio-dependent predator–prey system with constant rate harvesting.  SIAM Journal on Applied Mathematics.  65 (3), 737–753 (2005).
  11. Xiao D., Li W., Han M.  Dynamics in a ratio-dependent predator-prey model with predator harvesting.  Journal of Mathematical Analysis and Applications.  324 (1), 14–29 (2006).
  12. Zhang Y., Zhang Q.  Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting.  Nonlinear Dynamics.  66 (1), 231–245 (2011).
  13. Seo G., Kot M.  A comparison of two predator-prey models with Holling's type I functional response.  Mathematical Biosciences.  212 (2), 161–179 (2008).
  14. Ang T. K., Safuan H. M., Kavikumar J.  The impact of harvesting activities on prey–predator fishery model in the presence of toxin.  Jounal of Science and Technology.  10 (2), 128–135 (2018).
  15. Chauhan S., Bhatia S. K., Chaudhary P.  Effect of pollution on prey–predator system with infected predator.  Communication in Mathematical Biology and Neuroscience. 14 (2017).
  16. Kumar U., Mandal P. S.  Role of Allee effect on prey–predator model with component Allee effect for predator reproduction.  Mathematics and Computers in Simulation.  193, 623–665 (2022).
  17. Zhang H., Cai Y., Fu S., Wang W.  Impact of the fear effect in a prey–predator model incorporating a prey refuge.  Applied Mathematics and Computation.  356, 328–337 (2019).
  18. Bohn J., Rebaza J., Speer K.  Continuous threshold prey harvesting in predator–prey models.  International Journal of Mathematical and Computational Sciences.  5 (7), 996–1003 (2011).
  19. Su J.  Degenerate Hopf bifurcation in a Leslie–Gower predator–prey model with predator harvest.  Advances in Difference Equations.  2020, 194 (2020).
  20. Dai G., Tang M.  Coexistence region and global dynamics of a harvested predator–prey system.  Journal on Applied Mathematics.  58 (1), 193–210 (1998).
  21. Haque M., Sarwardi S.  Effect of toxicity on a harvested fishery model.  Modeling Earth Systems and Environment.  2 (3), 122 (2016).
  22. Vijayalakshmi T., Senthamarai R.  Study of two species prey–predator model in imprecise environment with harvesting scenario.  Mathematical Modeling and Computing.  9 (2), 385–398 (2022).
  23. Birkoff G., Rota G. C.  Ordinary differential equations. New York, Wiley (1978).
  24. Hale J. K., Lunel S. M. V.  Introduction to Functional Differential Equations.  Springer Science and Business Media (2013).
  25. Dumortier F., Llibre J., Artés J. C.  Qualitative theory of planar differential systems.  Berlin, Springer (2006).
  26. Perko L. Differential equations and dynamical systems.  Springer–Verlag (2000).
  27. Rasedee A. F. N., Abdul Sathar M. H., Mohd Najib N., Wong T. J., Koo L. F.  Numerical analysis on chaos attractors using a backward difference formulation.  Mathematical Modeling and Computing.  9 (4), 898–908 (2022).