Periodic solutions for an equation governing dynamics of a renewable resource subjected to Allee effects. (English) Zbl 1197.34078

Summary: The minimum number of positive periodic solutions admitted by a non-autonomous scalar differential equation is estimated. This result is employed to find the minimum number of positive periodic solutions admitted by a model describing the dynamics of a renewable resource that is subjected to Allee effects in a seasonally varying environment. The Allee effect refers to a decrease in the population growth rate at low population densities. The Leggett-Williams multiple fixed point theorem is used to establish the existence of positive periodic solutions.


34C60 Qualitative investigation and simulation of ordinary differential equation models
34C25 Periodic solutions to ordinary differential equations
92D40 Ecology
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI


[1] Cheng, S.; Zhang, G., Existence of positive periodic solutions for non autonomous functional differential equations, Electron. J. Differential Equations, 59, 1-8 (2001)
[2] Jiang, D.; Wei, J.; Jhang, B., Positive periodic solutions of functional differntial equations and population models, Electron. J. Differential Equations, 71, 1-13 (2002)
[3] Wan, A.; Jiang, D., existence of positive periodic solutions for fucntional differential equations, Kyushu J. Math., 56, 193-202 (2002) · Zbl 1012.34068
[4] Wan, A.; Jiang, D., A new existence theory for positive periodic solutions to fucntional differential equations, Comput. Math. Appl., 47, 1257-1262 (2004) · Zbl 1073.34082
[5] Zhang, G.; Cheng, S., Positive periodic solutions of nonautonomous functional differential equations depending on a parameter, Abstr. Appl. Math., 7, 279-286 (2002) · Zbl 1007.34066
[6] Zhang, W.; Zhu, D.; Bi, Ping, Existence of periodic solutions of a scalar functional differential equations via a fixed point theorem, Math. Comput. Modelling, 46, 718-729 (2007) · Zbl 1145.34041
[7] Padhi, Seshadev; Srivastava, Shilpee, Multiple periodic solutions for nonlinear first order functional differential equations with appliations to population dynamics, Appl. Math. Comput., 203, 1, 1-6 (2008) · Zbl 1161.34349
[8] Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033
[9] Barnett, A., Safety in numbers, New Sci., 169, 38-41 (2001)
[10] Fowler, C. W.; Baker, J. D., A review of animal populations dynamics at extremely reduced population levels, Rep. Int. Whal. Comm., 41, 545-554 (1991)
[11] Stephens, P. A.; Sutherland, W. J.; Freckleton, R. P., What is the Allee effect?, Oikos, 87, 185-190 (1999)
[12] Begon, M.; Harper, J. L.; Townsend, C. R., Ecology, Individuals, Populations and Communities (1996), Blackwell Science: Blackwell Science Oxford
[13] Burgman, M. A.; Ferson, S.; Akcakaya, H. R., Risk Assessment in Conservation Biology (1993), Chapman and Hall: Chapman and Hall London
[14] Courchamp, Franck; Clutton-Brock, Tim; Grenfell, Bryan, Inverse density dependence and the Allee effect, TREE, 14, 10, 405-410 (1999)
[15] Fretwell, S. D., Populations in a Seasonal Environment (1972), Princeton Univ. Press: Princeton Univ. Press Princeton, NY
[16] May, R. M., Stability and Complexity in Model Ecosystems (1973), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ
[17] Odum, E. P., Fundamentals of Ecology (1959), Saunders: Saunders Philadelphia
[18] Sther, B. E.; Ringsby, T. H.; Rskaft, E., Life history variation, population processes and priorities in species conservation: Towards a reunion of research paradigms, Oikos, 77, 217-226 (1996)
[19] Berec, Luděk; Angulo, Elena; Courchamp, Franck, Multiple Allee effects and population management, TREE, 22, 4, 185-191 (2006)
[20] Hurford, Amy; Hebblewhite, Mark; Lewis, M. A., A spatially explicit model for an Allee effect: Why wolves recolonize so slowly in Greater Yellowstone, Theoret. Popul. Biol., 70, 244-254 (2006) · Zbl 1112.92045
[21] Brockett, Beth F. T.; Hassall, Mark, The existence of an Allee effect in populations of Porcellio scaber (Isopoda: Oniscidea), Eur. J. Soil Biol., 41, 123-127 (2005)
[22] Hilker, Frank; Langlais, Michael; Petrovskii, Sergei V.; Malchow, Horst, A diffusive SI model with Allee effect and application to FIV, Math. Biosci., 61-80 (2007) · Zbl 1124.92044
[23] Gardner, Janet L., Winter flocking behaviour of speckled warblers and the Allee effect, Biol. Conserv., 118, 195-204 (2004)
[24] Kussaari, Mikko; Saccheri, Ilik; Camara, Mark; Hanski, Ilkka, Allee effect and population dynamics in the Glanville fritillary butterfly, Oikos, 82, 2, 384-392 (1998)
[25] Penteriani, Vincenzo; Otalora, Fermín; Ferrer, Miguel, Floater mortality within settlement areas can explain the Allee effect in breeding populations, Ecol. Model., 213, 98-104 (2007)
[26] Boukal, David S.; Berec, Luděk, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theoret. Biol., 218, 375-394 (2002)
[27] Clark, Colin W., Mathematical Bioeconomics - The Optimal Management of Renewable Resources (2005), Wiley-Interscience · Zbl 0364.90002
[28] Kot, Mark, Elements of Methematical Ecology (2001), Cambridge University Press: Cambridge University Press Cambridge
[29] Castilho, César; Srinivasu, Pichika D. N., Bio-economics of a renewable resource in a seasonally varying environment, Math. Biosc., 205, 1-18 (2007) · Zbl 1106.92063
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.