Hopf bifurcation in a spatial predator-prey model with the square root functional response for the predator

Document Type : Original Paper

Authors

1 Department of Mathematical Sciences, University of Kashan, Kashan, Iran, 87317-53153

2 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran, 84156-83111

Abstract

‎In this paper‎, ‎we consider a diffusive predator-prey model‎, ‎in which the prey population lives in groups and has a social behavior‎. ‎We show that Hopf bifurcation and the existence of a center manifold may occur‎.

‎The linear stability analysis shows that a Hopf bifurcation occurs in the corresponding homogeneous system‎.

‎Next‎, ‎we study the effect of diffusion parameters on homogeneous dynamics‎.

‎By choosing a proper bifurcation parameter‎, ‎we prove that a Hopf bifurcation occurs in the nonhomogeneous system‎. ‎We compute the normal form of this bifurcation up to the third order and obtain the direction of the Hopf bifurcation‎. ‎Finally‎, ‎we provide numerical simulations to illustrate our analytical findings‎.

Highlights

 

Adams R. A., Sobolev spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York, London, 197

. Adams R.A. and Fournier J.J.F., Sobolev spaces, London: Academic Press, 200

Ajraldi V. and Venturino E., Mimicking spatial effects in predator-prey models with group defense, Proceedings of the 2009 International Conference on Computational and Mathematical Methods in Science and Engineering, 1 (2009) 57-67.

Ajraldi V., Pittavino M. and Venturino E., Modeling Herd behavior in population systems, Nonlin. Anal. Real World App., 12 (2011) 2319-2338.

Boudjema I. and Djilali S., Turing-Hopf bifurcation in Gauss-type model with cross diffusion and its application, Nonlinear Stud., 25 (2018) 665-687.

 Braza P.A., Predator-prey dynamics with square root functional responses, Nonlin. Anal. Real World. Appl., 13 (2012) 1837-1843.

 Cosner C., DeAngelis D.L., Ault J.S. and Olson D.B., Effects of spatial grouping on the functional response of predators, Theoretical, Population Biology, 56 (1999) 65-75.

Djilali S., Herd behavior in a predator-prey model with spatial diffusion: bifurcation analysis and Turing instability, J. Appl. Math. Comput., 58 (2018) 125-149

Djilali S., Impact of prey herd shape on the predator-prey interaction, Chaos, Solitons and Fractals, 120 (2019) 139-148

Haragus M. and Iooss G., Local Bifurcations, Center Manifolds, and Normal Forms in InfiniteDimensional Dynamical Systems, Universitext, Springer London, 2011.

Keywords

Main Subjects


 
Adams R. A., Sobolev spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York, London, 197
. Adams R.A. and Fournier J.J.F., Sobolev spaces, London: Academic Press, 200
Ajraldi V. and Venturino E., Mimicking spatial effects in predator-prey models with group defense, Proceedings of the 2009 International Conference on Computational and Mathematical Methods in Science and Engineering, 1 (2009) 57-67.
Ajraldi V., Pittavino M. and Venturino E., Modeling Herd behavior in population systems, Nonlin. Anal. Real World App., 12 (2011) 2319-2338.
Boudjema I. and Djilali S., Turing-Hopf bifurcation in Gauss-type model with cross diffusion and its application, Nonlinear Stud., 25 (2018) 665-687.
 Braza P.A., Predator-prey dynamics with square root functional responses, Nonlin. Anal. Real World. Appl., 13 (2012) 1837-1843.
 Cosner C., DeAngelis D.L., Ault J.S. and Olson D.B., Effects of spatial grouping on the functional response of predators, Theoretical, Population Biology, 56 (1999) 65-75.
Djilali S., Herd behavior in a predator-prey model with spatial diffusion: bifurcation analysis and Turing instability, J. Appl. Math. Comput., 58 (2018) 125-149
Djilali S., Impact of prey herd shape on the predator-prey interaction, Chaos, Solitons and Fractals, 120 (2019) 139-148
Haragus M. and Iooss G., Local Bifurcations, Center Manifolds, and Normal Forms in InfiniteDimensional Dynamical Systems, Universitext, Springer London, 2011.