The paper deals with a non-autonomous Lotka–Volterra type system, which in particular may include logistic growth of the prey population and hunting cooperation between predators. We focus on the existence of positive periodic solutions by using an operator approach based on the Krasnosel’skii homotopy expansion theorem. We give sufficient conditions in order that the localized periodic solution does not reduce to a steady state. Particularly, two typical expressions for the functional response of predators are discussed.
Universidade de Santiago de Compostela, Santiago de Compostela, Spain
Babes-Bolyai University, Cluj-Napoca, Romania
C. Lois-Parados, R. Precup, Positive periodic solutions for Lotka-Volterra systems with a general attack rate, Nonlinear Anal. Real World Appl. 52 (2020), pp 17, https://doi.org/10.1016/j.nonrwa.2019.103024
Nonlinear Analysis: Real World Applications
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1] M. Teixeira Alves, F.M. Hilker, Hunting cooperation and Allee effects in predators, J. Theoret. Biol. 419 (2017) 13–22.
 F. Brauer, C. Castillo-Ch ́avez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001.
 F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math. 21
 A.J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925.
 V. Volterra, Variazioni e fluttazioni del numero d’individui in specie animali conviventi, Mem. Acad. Sci. Lincei 2 (1926)
 H. Baca ̈er, A Short History of Mathematical Population Dynamics: Lotka–Volterra and the Predator–Prey System
(1920–1926), Springer, London, 2011, pp. 71–76.
 M.L. Rosenzweig, R.H. MacArthur, Graphical representation and stability conditions of predator–prey interactions,
Amer. Nat. 97 (1963) 209–223.
 Q. van der Hoff, T.H. Fay, A predator–prey model with predator population saturation, Math. Stat. 4 (2016) 101–107.
 G. Buffoni, M. Groppi, C. Soresina, Effects of prey over-undercrowding in predator–prey systems with prey-dependent
trophic function, Nonlinear Anal. RWA 12 (2011) 2871–2887.
 L. Berec, Impacts of foraging facilitation among predators on predator–prey dynamics, Bull. Math. Biol. 72 (2010)
 D.P. Tsvetkov, A periodic Lotka–Volterra system, Serdica Math. J. 22 (1996) 109–116.
 R. Precup, A vector version of Krasnosel’skii’s fixed point theorem in cones and positive periodic solutions of nonlinear
systems, J. Fixed Point Theory Appl. 2 (2007) 141–151.
 D. O’Regan, R. Precup, Theorem of Leray–Schauder Type and Applications, Gordon and Breach, Singapore, 2001.
 M.A. Krasnosel’skii, Positive Solutions of Operator Equations, P. Noordhoff Ltd, The Netherlands, 1964.