# zbMATH — the first resource for mathematics

Environmental periodicity and time delays in a “food-limited” population model. (English) Zbl 0701.92021
Summary: Sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the food-limited population system modelled by the equation $\dot N(t)=r(t)((K(t)-N(t- m\omega))/(K(t)+c(t)r(t)N(t-m\omega))),$ where m is a nonnegative integer and K,r,c are continuous, positive, periodic functions of period $$\omega$$.

##### MSC:
 92D40 Ecology 34C25 Periodic solutions to ordinary differential equations 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Arrigoni, M.; Steiner, A., Logistic growth in a fluctuating environment, J. math. biol., 21, 237-241, (1985) · Zbl 0563.92014 [2] Barbalat, I., Systémes d’équations différentielles d’oscillations nonlinéares, Rev. roumaine math. pures appl., 4, 267-270, (1959) · Zbl 0090.06601 [3] Boyce, M.S.; Daley, D.J., Population tracking of fluctuating environments and natural selection for tracking ability, Amer. natur., 115, 480-491, (1980) [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.; 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 [6] Emmel, T.C., Population biology, (1976), Harper & Row New York [7] Fleming, T.H.; Hooker, R.S., anolis cupreus: the response of a lizard to tropical seasonality, Ecology, 56, 1243-1261, (1975) [8] Fretwell, S.D., Populations in a seasonal environment, (1972), Princeton Univ. Press Princeton, NJ [9] Gopalsamy, K.; Kulenović, M.R.S.; Ladas, G., Time lags in a “food-limited” population model, Appl. anal., 31, 225-237, (1988) · Zbl 0639.34070 [10] Hallam, T.G.; DeLuna, J.T., Effects of toxicants on populations: A qualitative approach III. environmental and food chain pathways, J. theor. biol., 109, 411-429, (1984) [11] Nicholson, A.J., An outline of the dynamics of animal populations, Austral. J. zool., 2, 9-65, (1954) [12] Nisbet, R.M.; Gurney, W.S.C., Population dynamics in a periodically varying environment, J. theor. biol., 56, 459-475, (1976) [13] Pianka, E.R., Evolutionary ecology, (1974), Harper & Row New York [14] Pielou, E.C., An introduction to mathematical ecology, (1969), Wiley New York · Zbl 0259.92001 [15] Rosen, G., Time delays produced by essential nonlinearity in population growth models, Bull. math. biol., 28, 253-256, (1987) · Zbl 0614.92015 [16] Smith, F.E., Population dynamics in daphnia magna, Ecology, 44, 651-663, (1963) [17] Sonneveld, P.; Van Kan, J., On a conjecture about the periodic solution of the logistic equation, J. math. biol., 8, 285-289, (1979) · Zbl 0412.92019
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.