×

zbMATH — the first resource for mathematics

Sharp seasonal threshold property for cooperative population dynamics with concave nonlinearities. (English) Zbl 1398.37093
Summary: We consider a biological population whose environment varies periodically in time, exhibiting two very different “seasons”: one is favorable and the other one is unfavorable. For monotone differential models with concave nonlinearities, we address the following question: the system’s period being fixed, under what conditions does there exist a critical duration for the unfavorable season? By “critical duration” we mean that above some threshold, the population cannot sustain and extincts, while below this threshold, the system converges to a unique periodic and positive solution. We term this a “sharp seasonal threshold property” (SSTP, for short).
Building upon a previous result, we obtain sufficient conditions for SSTP in any dimension and apply our criterion to a two-dimensional model featuring juvenile and adult populations of insects.
MSC:
37N25 Dynamical systems in biology
34D23 Global stability of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aronsson, G.; Mellander, I., A deterministic model in biomathematics. asymptotic behavior and threshold conditions, Math. Biosci., 49, 207-222, (1980) · Zbl 0433.92025
[2] Bacaër, N.; Ait Dads, N., Sur l’interprétation biologique d’une définition du paramètre R0 pour LES modèles périodiques de populations, J. Math. Biol., 65, 601-621, (2012), (French)
[3] Bacaër, N., Sur le modèle stochastique SIS pour une épidémie dans un environnement périodique, J. Math. Biol., 71, 491-511, (2015), (French)
[4] Campillo, F.; Champagnat, N.; Fritsch, C., On the variations of the principal eigenvalue with respect to a parameter in growth-fragmentation models, Commun. Math. Sci., 15, 7, 1801-1819, (2017) · Zbl 1387.35585
[5] Clairambault, J.; Gaubert, S.; Perthame, B., An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations, C. R. Math., 345, 10, 549-554, (2007) · Zbl 1141.34326
[6] Gaubert, S.; Lepoutre, T., Discrete limit and monotonicity properties of the Floquet eigenvalue in an age structured cell division cycle model, J. Math. Biol., 71, 6, 1663-1703, (2015) · Zbl 1359.37148
[7] Hirsch, M. W., The dynamical systems approach to differential equations, Bull. Am. Math. Soc., 11, 1-64, (1984) · Zbl 0541.34026
[8] Kingman, J. F.C., A convexity property of positive matrices, Q. J. Math., 12, 283-284, (1961) · Zbl 0101.25302
[9] Smith, H. L., Cooperative systems of differential equations with concave nonlinearities, Nonlinear Anal., Theory Methods Appl., 10, 10, 1037-1052, (1986) · Zbl 0612.34035
[10] Jiang, J., The algebraic criteria for the asymptotic behavior of cooperative systems with concave nonlinearities, Syst. Sci. Math. Sci., 6, 3, 193-208, (1993) · Zbl 0787.34045
[11] Mirrahimi, S.; Perthame, B.; Souganidis, P., Time fluctuations in a population model of adaptive dynamics, Ann. Inst. Henri Poincaré (C) Non Linéaire Anal., 32, 1, 41-58, (2015) · Zbl 1312.35011
[12] Xiao, D., Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting, Discrete Contin. Dyn. Syst., Ser. B, 21, 2, 699-719, (2016) · Zbl 1331.34101
[13] Zhang, Z.; Ding, T.; Huang, W.; Dong, Z., Qualitative theory of differential equations, Translations of Mathematical Monographs, vol. 101, (1991), Amer. Math. Soc. Providence
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.