Abstract
In this paper, the schooling behavior of prey fish population in a predator–prey interaction is investigated. By taking an economical interest which can be elaborated by the presence of nonselective harvesting into consideration, we studied the dynamical behavior. The existence, positivity and boundedness of solution have been established. The analysis of the equilibrium states is presented by studying the local and the global stability. The possible types of local bifurcation that the system can undergoes are discussed. The effect of fishing effort on the evolution of the species is examined. Further, by using Pontryagin’s maximum principle a proper management strategy has been used for avoiding the extinction of the considered species and maximizing the benefits. For the validation of the theoretical result, several of graphical representations have been used.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ajraldi V, Pittavino M, Venturino E (2011) Modelling herd behaviour in population systems. Nonlinear Anal Real Worl Appl 12(4):2319–2338
Banerjee M, Petrovskii S (2011) Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system. Theor Ecol 4(1):37–53
Belkhodja K, Mousaoui A, Alaoui MAA (2018) Optimal harvesting and stability for a prey-predator model. Nonlinear Anal Real World Appl 39:321–336
Boudjema I, Djilali S (2018) Turing-Hopf bifurcation in Gauss-type model with cross diffusion and its application. Nonlinear Stud 25(3):665–687
Braza AP (2012) Predator-prey dynamics with square root functional responses. Nonlinear Anal Real Worl Appl 13:1837–1843
Cagliero E, Venturino E (2016) Ecoepidemics with infected prey in herd defense: the harmless and toxic cases. Int J Comput Math 93:108–127
Chow SN, Hale Jk (1982) Methods of bifurcation theory. Springer, New York
Clark CW (1990) Mathematical bioeconomics, the optimal management of renewable resources, 2nd edn. Wiley, New York
Djilali S (2018) Herd behaviour in a predator-prey model with spatial diffusion: bifurcation analysis and turing instability. J Appl Math Comput 58:125–149
Djilali S (2019) Impact of prey herd shape on the predator-prey interaction. Chaos, Solitons and Fractals 120:139–148
Djilali S (2019) Effect of herd shape in a diffusive predator-prey model with time delay. J Appl Anal Comput 9(2):638–654
Djilali S, Bentout S (2019) Spatiotemporal patterns in a diffusive predator-prey model with prey social behavior. Acta Applicandae Mathematicae 169:125–143
Ghosh B, Kar TK (2014) Sustainable use of prey species in a preypredator system: jointly determined ecological thresholds and economic trade-offs. Ecol Model 272:49–58
Hale J (1988) Ordinary differential equations. Krieger Malabar FL Appl 198:355–370
Lotka A (1925) Elements of physical biology. Williams and Wilkins, Baltimore
Leitmann G (1966) An introduction to optimal control. Mc Graw-Hill, New York
Murray JD (1989) Mathematical biology. Springer, New York
Pontryagin LS, Boltyonskii VG, Gamkrelidre RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, NewYork
Song Y, Yin T, Shu H (2017) Dynamics of ratio-dependent stage structured predator-prey model with delay. Math Method Appl Sci. https://doi.org/10.1002/mma.4467
Song Y, Tang X (2017) Stability, stady-state bifurcation, and turing patterns in predator-prey model with herd behavior and prey-taxis. Stud Appl Math 139(3):391–404
Song Y, Zou XF (2014) Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a Hopf-turing bifurcation point. Comput Math Appl 67:1978–1997
Song Y, Zou XF (2014) Bifurcation analysis of a diffusive ratio-dependent predator-prey model. Nonlinear Dyn 78:49–70
Solow RM (1974) The economics of resources or the resources of economics. Am Econ Rev 64:1–14
Tripathi JP, Tyagi S, Abbas S (2016) Global analysis of a delayed density dependent predator-prey model with Crowley-Martin functional response. Commun Nonlinear Sci Numer Simul 30(1):45–69
Venturino E (2011) A minimal model for ecoepidemics with group defense. J Biol Syst 19:763–785
Venturino E (2013) Modeling herd behavior in population systems. Nonlinear Anal Real World Appl 12(4):2319–2338
Venturino E, Petrovskii S (2013) Spatiotemporal behavior of a prey-predator system with a group defense for prey. Ecol Compl 14:37–47
Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos. Springer, New York
Xu Z, Song Y (2015) Bifurcation analysis of a diffusive predator-prey system with a herd behavior and quadratic mortality. Math Meth Appl Sci 38(4):2994–3006
Yang R, Wei J (2015) Stability and bifurcation analysis of a diffusive prey-predator system in holling type III with a prey refuge. Nonlinear Dyn 79:631–646
Zhang Y, Sanling Chaoqun X, Zhang T (2013) Spatial dynamics in a predator-prey model with herd behavior. Chaos: an interdisciplinary. J Nonlinear Sci 23(3):033102
Zhang G, Shen Y (2015) Periodic solutions for a neutral delay Hassell-Varley type predator-prey system. Appl Math Comput 264:443–52
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hacini, M.E.M., Hammoudi, D., Djilali, S. et al. Optimal harvesting and stability of a predator–prey model for fish populations with schooling behavior. Theory Biosci. 140, 225–239 (2021). https://doi.org/10.1007/s12064-021-00347-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12064-021-00347-5