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Multiple attractors and resonance in periodically forced population models. (English) Zbl 0957.37018
Discrete autonomous dynamical systems with periodic solutions admit multiple oscillatory solutions in the advent of periodic forcing. In general, the multiple cycles are mutually out of phase, and some of the cycle averages may increase with the forcing amplitude while others decrease. The multiple cycles differ in phase, and may differ in average total population size as well. The author shows that the average total population size may resonate or attenuate with the amplitude of the environmental fluctuation depending on the initial population size. The author applies his result to the periodic LPA model, Tribolium populations were maintained in periodically forced flour habitats of constant \(20g\), alternating \(2s-12g\), and alternating \(32-12g\).

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37N25 Dynamical systems in biology
Full Text: DOI
[1] Smith, H.L., Competitive coexistence in an oscillating chemostat, SIAM J. appl. math., 40, 498-522, (1981) · Zbl 0467.92018
[2] Jillson, D., Insect populations respond to fluctuating environments, Nature, 288, 699-700, (1980)
[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] Costantino, R.F.; Cushing, J.M.; Dennis, B.; Desharnais, R.A.; Henson, S.M., Resonant population cycles in alternating habitats, Bull. math. biol., 60, 247-273, (1998) · Zbl 0973.92034
[5] S.M. Henson, R.F. Costantino, J.M. Cushing, B. Dennis, R.A. Desharnais, Multiple attractors, saddles, and population dynamics in periodic habitats, Bull. Math. Biol. 61 (1999) 1121-1149. · Zbl 1323.92169
[6] Dennis, B.; Desharnais, R.A.; Cushing, J.M.; Costantino, R.F., Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments, Ecol. mono., 65, 261-281, (1995)
[7] Dennis, B.; Desharnais, R.A.; Cushing, J.M.; Costantino, R.F., Transitions in population dynamics: equilibria to periodic cycles to aperiodic cycles, J. anim. ecol., 66, 704-729, (1997)
[8] Costantino, R.F.; Cushing, J.M.; Dennis, B.; Desharnais, R.A., Experimentally induced transitions in the dynamic behavior of insect populations, Nature, 375, 227-230, (1995)
[9] Costantino, R.F.; Desharnais, R.A.; Cushing, J.M.; Dennis, B., Chaotic dynamics in an insect population, Science, 275, 389-391, (1997) · Zbl 1225.37103
[10] K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, New York, 1997, pp. 71 and 558.
[11] J.P. LaSalle, The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics 25, SIAM, Philadelphia, PA, 1976. · Zbl 0364.93002
[12] Henson, S.M., The effect of periodicity in maps, J. difference eq. appl., 5, 31-56, (1999) · Zbl 0988.39011
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