Uniqueness of periodic solutions of a nonautonomous density-dependent predator–prey system

Abstract In this present paper, we investigate the uniqueness of periodic solutions of a nonautonomous density-dependent and ratio-dependent predator–prey system, where not only the prey density dependence but also the predator density dependence are considered, such that the studied predator–prey system conforms to the realistically biological environment. We start with a sufficient condition for the permanence of the system and then construct a weaker sufficient condition by introducing a specific set, denoted as Γ . Based on this Γ and the Brouwer fixed-point theorem, we obtain the existence condition of positive periodic solutions. Moreover, since the uniqueness of positive periodic solutions can be ensured by global attractiveness, we alternatively introduce a sufficient condition for global attractiveness. Similarly, we also provide a sufficient condition for the uniqueness of non-negative periodic solutions.

[1]  R. Arditi,et al.  Functional responses and heterogeneities: an experimental test with cladocerans , 1991 .

[2]  Jiandong Zhao,et al.  Permanence in nonautonomous Lotka-Volterra system with predator-prey , 2004, Appl. Math. Comput..

[3]  Shengqiang Liu,et al.  A Stage-structured Predator-prey Model of Beddington-DeAngelis Type , 2006, SIAM J. Appl. Math..

[4]  D. Bainov,et al.  Impulsive Differential Equations: Periodic Solutions and Applications , 1993 .

[5]  Haiyin Li,et al.  Dynamics of the density dependent predator–prey system with Beddington–DeAngelis functional response , 2011 .

[6]  Jing'an Cui.  Permanence of predator-prey system with periodic coefficients , 2005, Math. Comput. Model..

[7]  R. Arditi,et al.  From pattern to process: identifying predator–prey models from time-series data , 2001, Population Ecology.

[8]  Yasuhiro Takeuchi,et al.  Permanence, extinction and periodic solution of predator–prey system with Beddington–DeAngelis functional response , 2006 .

[9]  Jim M Cushing,et al.  Periodic Time-Dependent Predator-Prey Systems , 1977 .

[10]  R. Arditi,et al.  Underestimation of mutual interference of predators , 1990, Oecologia.

[11]  S. Kakutani.  A generalization of Brouwer’s fixed point theorem , 1941 .

[12]  Mark A. J. Chaplain,et al.  Persistence and attractivity in an N-species ratio-dependent predator-prey system with distributed time delays , 2002, Appl. Math. Comput..

[13]  Yang Kuang,et al.  Global qualitative analysis of a ratio-dependent predator–prey system , 1998 .

[14]  Qian Wang,et al.  Dynamics of a non-autonomous ratio-dependent predator—prey system , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[15]  Huaiping Zhu,et al.  Bifurcation Analysis of a Predator-Prey System with Nonmonotonic Functional Response , 2003, SIAM J. Appl. Math..

[16]  Matthijs Vos,et al.  Functional responses modified by predator density , 2009, Oecologia.

[17]  Paul M. Dolman,et al.  The intensity of interference varies with resource density: evidence from a field study with snow buntings, Plectrophenax nivalis , 1995, Oecologia.

[18]  Zhikun She,et al.  Dynamics of a density-dependent stage-structured predator–prey system with Beddington–DeAngelis functional response , 2013 .

[19]  Zhidong Teng,et al.  Persistence in nonautonomous predator-prey systems with infinite delays , 2006 .

[20]  J. F. Gilliam,et al.  FUNCTIONAL RESPONSES WITH PREDATOR INTERFERENCE: VIABLE ALTERNATIVES TO THE HOLLING TYPE II MODEL , 2001 .

[21]  Ke Wang,et al.  Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses ✩ , 2003 .

[22]  K. R. Schneider,et al.  Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations , 2013 .

[23]  R. Arditi,et al.  Coupling in predator-prey dynamics: Ratio-Dependence , 1989 .

[24]  D. Baĭnov,et al.  Systems with impulse effect : stability, theory, and applications , 1989 .

[25]  Yang Kuang,et al.  Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response , 2004 .

[26]  Jianhua Wu,et al.  Qualitative analysis for a ratio-dependent predator–prey model with stage structure and diffusion☆ , 2008 .

[27]  Wonlyul Ko,et al.  A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: I, long time behavior and stability of equilibria , 2013 .