Effect of periodic environmental fluctuations on the Pearl-Verhulst model. (English) Zbl 1197.34062

Summary: We address the effect of periodic environmental fluctuations on the Pearl-Verhulst model in population dynamics and clarify several important issues very actively discussed in recent papers of B. S. Lakshmi [ibid. 16, 183–186 (2003); ibid. 26, 719–721 (2005; Zbl 1032.92028)], P. G. L. Leach and K. Andriopoulos [ ibid. 22, 1183–1188 (2004; Zbl 1081.92033)], J. H. Swart and H. C. Murrell [ibid. 32, 1325–1327 (2007; Zbl 1195.92054)]. Firstly, we review general results regarding existence and properties of periodic solutions and examine existence of a unique positive asymptotically stable periodic solution of a non-autonomous logistic differential equation when \(r(t)>0\). Proceeding to the case where \(r(t)\) is allowed to take on negative values, we consider a modified Pearl-Verhulst equation because, as emphasized by T. G. Hallam and C. E. Clark [J. Theor. Biol. 93, 303–311 (1981)], use of the classic one leads to paradoxical biological conclusions. For a modified logistic equation with \(\omega \)-periodic coefficients, we establish existence of a unique asymptotically stable positive periodic solution with the same period. Special attention is paid to important cases where time average of the intrinsic growth rate is non-positive. Results of computer simulation demonstrating advantages of a modified equation for modeling periodic environmental fluctuations are presented.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.


34C25 Periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
Full Text: DOI


[1] Aiello, W., The existence of nonoscillatory solutions to a generalized, nonautonomous, delay logistic equation, J Math Anal Appl, 149, 114-123 (1990) · Zbl 0711.34091
[2] Arrigoni, M.; Steiner, A., Logistiches Wachstum in fluktuierender Umwelt, Math Biol, 21, 237-241 (1985) · Zbl 0563.92014
[3] Brauer, F.; Sánchez, D. A., Periodic environments and periodic harvesting, Nat Resour Model, 16, 233-244 (2003) · Zbl 1067.92056
[4] Coleman, B. D., Nonautonomous logistic equations as models of the adjustment of populations to environmental change, Math. Biosci., 45, 159-173 (1979) · Zbl 0425.92013
[5] Coleman, B. D., On optimal intrinsic growth rates for populations in periodically changing environments, J Math Biol, 12, 343-354 (1981) · Zbl 0485.92017
[6] Coleman, B. D.; Hsieh, Y.-H.; Knowles, G. P., On the optimal choice of \(r\) for a population in a periodic environment, Math Biosci, 46, 71-85 (1979) · Zbl 0429.92022
[7] Curle, N., Applied differential equations (1972), Van Nostrand Reinhold: Van Nostrand Reinhold London New York Cincinnati Toronto Melbourne · Zbl 0243.34002
[8] Floquet, G., Sur les équations différentielles linéaires a coefficients périodiques, Annales École Normale, 13, 47-88 (1883) · JFM 15.0279.01
[9] Freedman, H. I.; Sree Hari Rao, V.; So, J. W.-H., Asymptotic behavior of a time-dependent single-species model, Analysis, 9, 217-223 (1989) · Zbl 0678.34056
[10] Freedman, H. I.; Wu, J., Periodic solutions of single-species models with periodic delay, SIAM J Math Anal, 23, 689-701 (1992) · Zbl 0764.92016
[11] Gopalsami, K.; Kulenovic, M. R.S.; Ladas, G., Environmental periodicity and time delays in a “food-limited” population model, J Math Anal Appl, 147, 545-555 (1990) · Zbl 0701.92021
[12] Hallam, T. G.; Clark, C. E., Non-autonomous logistic equations as models of populations in deteriorating environment, J Theor Biol, 93, 303-311 (1981)
[13] Jilson, D. A., Insect populations respond to fluctuating environments, Nature, 288, 699-700 (1980)
[14] Lakshmi, B. S., Oscillating population models, Chaos Solitons & Fractals, 16, 183-186 (2003) · Zbl 1032.92028
[15] Lakshmi, B. S., Population models with time dependent parameters, Chaos Solitons & Fractals, 26, 719-721 (2005) · Zbl 1081.92033
[16] Lazer, A. C.; Sánchez, D. A., Constant rate population harvesting: equilibrium and stability, Theor Pop Biol, 8, 12-30 (1975) · Zbl 0313.92012
[17] Lazer, A. C.; Sánchez, D. A., Periodic equilibria under periodic harvesting, Math Mag, 57, 156-158 (1984) · Zbl 0539.92026
[18] Leach, P. G.L.; Andriopoulos, K., An oscillatory population model, Chaos Solitons & Fractals, 22, 1183-1188 (2004) · Zbl 1063.92043
[19] Lopez-Ruiz, R.; Fournier-Prunaret, D., Indirect Allee effect, bistability and chaotic oscillations in a predator-prey discrete model of logistic type, Chaos Solitons & Fractals, 24, 85-101 (2005) · Zbl 1066.92053
[20] MacArtur, R.; Wilson, E. O., The theory of island biogeography (1973), Princeton University Press: Princeton University Press Princeton
[21] Neto, A. L., On the number of solutions of the equation \(\frac{d x}{d t} = \sum_{j = 0}^n a_j(t) x^j, 0 \leqslant t \leqslant 1\), for which \(x(0) = x(1)\), Invent Math, 59, 67-76 (1980)
[22] Nicholson, A. J., An outline of the dynamics of animal populations, Aust J Zool, 2, 9-65 (1954)
[23] Nisbet, R. M.; Gurney, W. S.C., Population dynamics in a periodically varying environment, J Theor Biol, 56, 459-475 (1976)
[24] Nisbet, R. M.; Gurney, W. S.C., Modelling fluctuating populations (1982), John Wiley & Sons: John Wiley & Sons Chichester - New York - Brisbane - Toronto - Singapore · Zbl 0593.92013
[25] Pliss, V. A., Nonlocal problems of the theory of oscillations (1966), Academic Press: Academic Press New York and London · Zbl 0151.12104
[26] Renshaw, E., Modelling biological populations in space and time (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0754.92018
[27] Sánchez, D. A., Periodic environments, harvesting, and a Riccati equation, (Laksmikantham, V., Nonlinear phenomena in mathematical sciences (1982), Academic Press: Academic Press New York) · Zbl 0582.34054
[28] Swart, J. H.; Murrell, H. C., An oscillatory model revisited, Chaos Solitons & Fractals, 32, 1325-1327 (2007) · Zbl 1195.92054
[29] Thieme, H. R., Mathematics in population biology (2003), Princeton University Press: Princeton University Press Princeton and Oxford · Zbl 1054.92042
[30] Vance, R. R.; Coddington, E. A., A nonautonomous model of population growth, J Math Biol, 27, 491-506 (1989) · Zbl 0716.92016
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.