×

zbMATH — the first resource for mathematics

A time-delay model of single-species growth with stage structure. (English) Zbl 0719.92017
The authors consider the global asymptotic stability of the positive equilibrium of a single-species growth model with stage structure consisting of immature and mature stages. Let \(x_ i(t)\) and \(x_ m(t)\) denote the concentration of immature and mature populations, respectively. We assume that the populations entering the environment over a time interval equal to the length of time from birth to maturity is \(\tau >0\). The model takes the form \[ \dot x_ i(t)=\alpha x_ m(t)-\gamma x_ i(t)-e^{-\gamma \tau}\phi (t-\tau),\quad \dot x_ m(t)=e^{-\gamma \tau}\phi (t-\tau)-\beta x^ 2_ m(t),\quad 0<t\leq \tau; \]
\[ \dot x_ i(t)=\alpha x_ m(t)-\gamma x_ i(t)-\alpha e^{-\gamma \tau}x_ m(t-\tau),\quad \dot x_ m(t)=\alpha e^{-\gamma \tau}x_ m(t-\tau)-\beta x^ 2_ m(t),\quad t>\tau, \] where \(\phi\) (t) is the birth rate of \(x_ i(t)\) at time t, -\(\tau\leq t\leq 0\), and \(\alpha,\beta,\gamma >0\) are constants. Oscillation and nonoscillation of solutions are addressed analytically and numerically. The effect of the delay on the population at equilibrium is also considered.

MSC:
92D25 Population dynamics (general)
34K20 Stability theory of functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aiello, W.G., The existence of nonoscillatory solutions to a generalized, nonautonomous, delay logistic equation, J. math. anal. appl., 149, 114-123, (1990) · Zbl 0711.34091
[2] Anderson, F.S., Competition in populations of one age group, Biometrica, 16, 19-27, (1960) · Zbl 0095.34001
[3] Barclay, H.J.; Van den Driessche, P., A model for a species with two life history stages and added mortality, Ecol. model., 11, 157-166, (1980)
[4] Beddington, J.R.; May, R.M., Time delays are not necessarily destabilizing, Math. biosci., 27, 109-117, (1975) · Zbl 0319.93037
[5] Blythe, S.P.; Nisbet, R.M.; Gurney, W.S.C., Instability and complex dynamics behaviour in population models with long time delays, Theor. pop. biol., 22, 147-176, (1982) · Zbl 0493.92019
[6] Bulmer, M.G., Periodical insects, Am. nat., 111, 1099-1117, (1977)
[7] Caswell, H., A simulation study of a time lag population model, J. theor. biol., 34, 419-439, (1972)
[8] Cunningham, W.J., A nonlinear differential-difference equation of growth, Proc. natl. acad. sci. USA, 40, 708-713, (1954) · Zbl 0055.31601
[9] De Angelis, D.L., Global asymptotic stability criteria for models of density-dependent population growth, J. theor. biol., 50, 35-43, (1975)
[10] Desharnais, R.A.; Liu, L., Stable demographic limit cycles in a laboratory populations of tribolium castaneum, J. animal ecol., 56, 885-906, (1987)
[11] Driver, R.D., Ordinary and delay differential equations, (1977), Springer New York · Zbl 0374.34001
[12] Fisher, M.E.; Goh, B.S., Stability results for delayed-recruitment models in population dynamics, J. math. biol., 19, 147-156, (1984) · Zbl 0533.92017
[13] Freedman, H.I.; Gopalsamy, K., Global stability in time-delayed single-species dynamics, Bull. math. biol., 48, 485-492, (1986) · Zbl 0606.92020
[14] Gurney, W.S.C.; Nisbet, R.M., Fluctuating periodicity, generation separation, and the expression of larval competition, Theor. pop. biol., 28, 150-180, (1985) · Zbl 0568.92018
[15] Gurney, W.S.C.; Blythe, S.P.; Nisbet, R.M., Nicholson’s blowflies revisited, Nature, 287, 17-21, (1980)
[16] Gurney, W.S.C.; Nisbet, R.M.; Lawton, J.H., The systematic formulation of tractable single species population models incorporating age structure, J. animal ecol., 52, 479-495, (1983)
[17] Hsieh, Y.-H., The phenomenon of unstable oscillation in population models, Math. comput. model., 10, 429-435, (1988) · Zbl 0647.92017
[18] Kakutani, S.; Markus, L., On the nonlinear difference-differential equation \(ẏ(t)=[A − By(t − τ)]y(t)\), (), 1-18 · Zbl 0082.30301
[19] Koslesov, Yu.S., Properties of solutions of a class of equations with lag which describe the dynamics of change in the population of a species with the age structure t aken into account, Math. USSR sb., 45, 91-100, (1983) · Zbl 0513.92017
[20] Kulenovic, M.R.S.; Ladas, G., Linearized oscillations in population dynamics, Bull. math. biol., 49, 615-627, (1987) · Zbl 0634.92013
[21] Landahl, H.D.; Hanson, B.D., A three stage population model with cannibalism, Bull. math. biol., 37, 11-17, (1975) · Zbl 0295.92015
[22] MacDonald, N., Time lags in biological models, () · Zbl 0403.92020
[23] May, R.M.; Conway, G.R.; Hassell, M.P.; Southwood, T.R.E., Time delays, density dependence, and single-species oscillations, J. animal ecol., 43, 747-770, (1974)
[24] Mazanov, A., On the differential-difference growth equation, Search, 4, 199-201, (1973)
[25] Rosen, G., Time delays produced by essential nonlinearity in population growth models, Bull. math. biol., 49, 253-255, (1987) · Zbl 0614.92015
[26] Ross, G.G., A difference-differential model in population dynamics, J. theor. biol., 34, 477-492, (1972)
[27] Stirzaker, D., On a population model, Math. biosci., 23, 329-336, (1975) · Zbl 0317.92024
[28] Tognetti, K., The two stage stochastic model, Math. biosci., 25, 195-204, (1975) · Zbl 0317.92026
[29] Wangersky, P.J.; Cunningham, W.J., On time lags in equations of growth, Proc. natl. acad. sci. USA, 42, 699-702, (1956) · Zbl 0072.37005
[30] Wood, S.N.; Blythe, S.P.; Gurney, W.S.C.; Nisbet, R.M., Instability in mortality estimation schemes related to stage-structure population models, IMA J. math. appl. med. biol., 6, 47-68, (1989) · Zbl 0659.92018
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.