×

Dynamics of a single species under periodic habitat fluctuations and Allee effect. (English) Zbl 1239.37013

The paper is concerned with the existence of periodic solutions to a first order nonautonomous nonlinear differential equation that models the dynamics of a single species under the influence of the Allee effect. Combining the direction field argument with the method of upper and lower solutions, the authors completely describe population dynamics in the cases of strong (\(A>0\)) and weak (\(A<0\)) Allee effect. They also provide bounds for periodic solutions and estimates for backward blow-up times. Remarkably, a recent general result due to S. Padhi, P. D. N. Srinivasu and G. K. Kumar [Nonlinear Anal., Real World Appl. 11, No. 4, 2610–2618 (2010; Zbl 1197.34078)] does not apply to the simple equation considered as an illustrative example in the paper under review.

MSC:

37N25 Dynamical systems in biology
92D25 Population dynamics (general)
34C25 Periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations

Citations:

Zbl 1197.34078
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Nicholson, A. J., An outline of the dynamics of animal populations, Aust. J. Zool., 2, 9-65 (1954)
[2] Vance, R. R.; Coddington, E. A., A nonautonomous model of population growth, J. Math. Biol., 27, 491-506 (1989) · Zbl 0716.92016
[3] Henson, S. M.; Cushing, J. M., The effect of periodic habitat fluctuations on a nonlinear insect population model, J. Math. Biol., 36, 201-226 (1997) · Zbl 0890.92023
[4] Brauer, F.; Sánchez, D. A., Periodic environments and periodic harvesting, Nat. Resour. Model., 16, 233-244 (2003) · Zbl 1067.92056
[5] Rogovchenko, S. P.; Rogovchenko, Yu. V., Effect of periodic environmental fluctuations on the Pearl-Verhulst model, Chaos Solitons Fractals, 39, 1169-1181 (2009) · Zbl 1197.34062
[6] Amarasekare, Priyanga, Allee effects in metapopulation dynamics, Am. Nat., 152, 298-302 (1998) · Zbl 0894.92029
[7] Lewis, M. A.; Kareiva, P., Allee dynamics and the spread of invading organisms, Theor. Popul. Biol., 43, 141-158 (1993) · Zbl 0769.92025
[8] Courchamp, F.; Clutton-Brock, T.; Grenfell, B., Inverse density dependence and the Allee effect, TREE, 14, 405-410 (1999)
[9] Courchamp, F.; Berec, L.; Gascoigne, J., Allee Effects in Ecology and Conservation (2008), Oxford University Press Inc.: Oxford University Press Inc. New York
[10] Boukal, D. S.; Berec, L., Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters, J. Theoret. Biol., 218, 375-394 (2002)
[11] Berec, L.; Angulo, E.; Courchamp, F., Multiple Allee effects and population management, TREE, 22, 185-191 (2006)
[12] Padhi, Seshadev; Srinivasu, P. D.N.; Kiran Kumar, G., Periodic solutions for an equation governing dynamics of a renewable resource subjected to Allee effects, Nonlinear Anal. RWA, 11, 2610-2618 (2010) · Zbl 1197.34078
[13] 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
[14] Langa, J. A.; Robinson, J. C.; Suárez, A., Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity, 15, 887-903 (2002) · Zbl 1004.37032
[15] Berger, A.; Siegmund, S., Uniformly attracting solutions of nonautonomous differential equations, Nonlinear Anal., 68, 3789-3811 (2008) · Zbl 1156.34042
[16] Hubbard, J. H.; West, B. H., Differential Equations: A Dynamical Systems Approach, Part I: Ordinary Differential Equations (1995), Springer-Verlag: Springer-Verlag New York
[17] Massera, J., The existence of periodic solutions of systems of differential equations, Duke Math. J., 17, 457-475 (1950) · Zbl 0038.25002
[18] Corduneanu, C., Principles of Differential and Integral Equations (1977), AMS Chelsea Publishing: AMS Chelsea Publishing The Bronx, New York · Zbl 0208.10701
[19] Hale, J. K.; Koçak, H., Dynamics and Bifurcations (1996), Springer-Verlag: Springer-Verlag New York
[20] Pliss, V. A., Nonlocal Problems of the Theory of Oscillations (1966), Academic Press: Academic Press New York, London · Zbl 0151.12104
[21] Friedrichs, K. O., Advanced Ordinary Differential Equations (1966), Gordon & Breach: Gordon & Breach New York · Zbl 0191.38202
[22] Sánchez, D. A., Ordinary Differential Equations: A Brief Eclectic Tour (2002), Mathematical Association of America: Mathematical Association of America Washington · Zbl 1004.34001
[23] Nkashama, M. N., A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations, J. Math. Anal. Appl., 140, 381-395 (1989) · Zbl 0674.34009
[24] Lakshmikantham, V.; Leela, S., Existence and monotone method for periodic solutions of first-order differential equations, J. Math. Anal. Appl., 159, 127-146 (1991)
[25] Andersen, K. M.; Sandqvist, A., On the number of closed solutions to an equation \(\dot{x} = f(t, x)\), where \(f_{x^n}(t, x) \geq 0 (n = 1, 2\), or \(3)\), J. Math. Anal. Appl., 194, 763-779 (1995)
[26] Korman, P.; Ouyang, Tiancheng, Exact multiplicity results for two classes of periodic equations, J. Math. Anal. Appl., 194, 763-779 (1995) · Zbl 0844.34036
[27] Lins Neto, A., On the number of solutions of the equation \(\frac{d x}{d t} = \sum_{j = 0}^n a_j(t) x^j, 0 \leq t \leq 1\), for which \(x(0) = x(1)\), Invent. Math., 59, 67-76 (1980) · Zbl 0448.34012
[28] Alvarez, C.; Lazer, A. C., An application of topological degree to the periodic competing species problem, J. Austral. Math. Soc. Ser. B, 28, 202-219 (1986) · Zbl 0625.92018
[29] Ahmad, S.; Montes de Oca, F., Extinction in nonautonomous \(T\)-periodic competitive Lotka-Volterra system, Appl. Math. Comput., 90, 155-166 (1998) · Zbl 0906.92024
[30] M. Hasanbulli, S.P. Rogovchenko, Yu.V. Rogovchenko, Dynamics of a single species in periodic environment under periodic harvesting, Preprint, 2010.; M. Hasanbulli, S.P. Rogovchenko, Yu.V. Rogovchenko, Dynamics of a single species in periodic environment under periodic harvesting, Preprint, 2010. · Zbl 1306.92044
[31] Bardi, M., An equation of growth of a single species with realistic dependence on crowding and seasonal factors, J. Math. Biol., 17, 33-43 (1983) · Zbl 0509.92021
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.