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Signature Function for Predicting Resonant and Attenuant Population 2-cycles

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Abstract

Populations are either enhanced via resonant cycles or suppressed via attenuant cycles by periodic environments. We develop a signature function for predicting the response of discretely reproducing populations to 2-periodic fluctuations of both a characteristic of the environment (carrying capacity), and a characteristic of the population (inherent growth rate). Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the carrying capacity and the demographic characteristic. Periodic environments are deleterious for populations when the signature function is negative. However, positive signature functions signal favorable environments. We compute the signature functions of six classical discrete-time single species population models, and use the functions to determine regions in parameter space that are either favorable or detrimental to the populations. The two-parameter classical models include the Ricker, Beverton-Holt, Logistic, and Maynard Smith models.

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References

  • 1 Begon, M., Harper, J.L., Townsend, C.R., 1996. Ecology: Individuals, populations and communities. Blackwell Science, Oxford.

    Google Scholar 

  • 2 Beverton, R.J.H., Holt, S.J., 1957. On the dynamics of exploited fish populations, H.M. Stationery Off., London. Fish. Invest. 2, 19.

    Google Scholar 

  • 3 Coleman, B.D., 1978. On the growth of populations with narrow spread in reproductive age. I. General theory and examples. J. Math. Biol. 6, 1–19.

    Article  MATH  MathSciNet  Google Scholar 

  • 4 Coleman, C.S., Frauenthal, J.C., 1983. Satiable egg eating predators, Math. Biosci. 63, 99–119.

    Article  MATH  MathSciNet  Google Scholar 

  • 5 Costantino, R.F., Cushing, J.M., Dennis, B., Desharnais, R. A., 1998. Resonant population cycles in temporarily fluctuating habitats. Bull. Math. Biol. 60, 247–273.

    Article  MATH  Google Scholar 

  • 6 Cull, P., 1986. Local and global stability for population models. Biol. Cybern. 54, 141–149.

    Article  MATH  MathSciNet  Google Scholar 

  • 7 Cull, P., 1988. Stability of discrete one-dimensional population models. Bull. Math. Biol. 50(1), 67–75.

    MATH  MathSciNet  Google Scholar 

  • 8 Cull, P., 2003. Stability in one-dimensional models. Sci. Math. Jpn. 58, 349–357.

    MathSciNet  Google Scholar 

  • 9 Cushing, J.M., Henson, S.M., 2001. Global dynamics of some periodically forced, monotone difference equations. J. Diff. Equ. Appl. 7, 859–872.

    MATH  MathSciNet  Google Scholar 

  • 11 Elaydi, S.N., 1994. Periodicity and stability of linear Volterra difference equations. J. Math. Anal. Appl. 181, 483–492.

    Article  MATH  MathSciNet  Google Scholar 

  • 10 Elaydi, S.N., 2000. Discrete Chaos. Chapman and Hall/CRC, Boca Raton, FL.

    MATH  Google Scholar 

  • 13 Elaydi, S.N., Sacker, R.J., 2005a. Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures. Proceedings of the Eighth International Conference on Difference Equations and Applications, 113–126, Chapman & Hall/CRC, Boca Raton, FL.

    Google Scholar 

  • 12 Elaydi, S.N., Sacker, R.J., 2005b. Global stability of periodic orbits of nonautonomous difference equations and population biology. J. Diff. Equ. 208(1), 258–273.

    Article  MATH  MathSciNet  Google Scholar 

  • 14 Elaydi, S.N., Sacker, R.J., 2005c. Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures. J. Difference Equ. Appl. 11(4–5), 337–346.

    Article  MATH  MathSciNet  Google Scholar 

  • 15 Elaydi, S.N., Sacker, R.J., in press. Periodic difference equations, populations biology and the Cushing-Henson conjectures (preprint).

  • 16 Elaydi, S.N., Yakubu, A.-A., 2002. Global stability of cycles: Lotka–Volterra competition model with stocking. J. Diff. Equ. Appl. 8(6), 537–549.

    MATH  MathSciNet  Google Scholar 

  • 17 Fisher, M.E., Goh, B.S., Vincent, T.L., 1979. Some stability conditions for discrete-time single species models. Bull. Math. Biol. 41, 861–875.

    MATH  MathSciNet  Google Scholar 

  • 18 Franke, J.E., Selgrade, J.F., 2003. Attractor for periodic dynamical systems. J. Math. Anal. Appl. 286, 64–79.

    Article  MATH  MathSciNet  Google Scholar 

  • 19 Franke, J.E., Yakubu, A.-A., 2005a. Periodic dynamical systems in unidirectional metapopulation models. J. Diff. Equ. Appl. 11(7), 687–700.

    MATH  MathSciNet  Google Scholar 

  • 20 Franke, J.E., Yakubu, A.-A. 2005b. Multiple attractors via cusp bifurcation in periodically varying environments. J. Diff. Equ. Appl. 11(4–5), 365–377.

    MATH  MathSciNet  Google Scholar 

  • 21 Franke, J.E., Yakubu, A.-A. 2005c. Population models with periodic recruitment functions and survival rates. J. Diff. Equ. Appl. 11(14), 1169–1184.

    MATH  MathSciNet  Google Scholar 

  • 22 Hassell, M.P., 1974. Density dependence in single species populations. J. Anim. Ecol. 44, 283- 296.

    Google Scholar 

  • 23 Hassell, M.P., Lawton, J.H., May, R.M., 1976. Patterns of dynamical behavior in single species populations. J. Anim. Ecol. 45, 471–486.

    Article  Google Scholar 

  • 25 Henson, S.M., 1999. The effect of periodicity in maps. J. Diff. Equ. Appl. 5, 31–56.

    Article  MATH  MathSciNet  Google Scholar 

  • 24 Henson, S.M., 2000. Multiple attractors and resonance in periodically forced population models. Phys. D 140, 33–49.

    Article  MATH  MathSciNet  Google Scholar 

  • 26 Henson, S.M., Costantino, R.F., Cushing, J.M., Dennis, B., Desharnais, R.A., 1999. Multiple attractors, saddles, and population dynamics in periodic habitats. Bull. Math. Biol. 61, 1121–1149.

    Article  Google Scholar 

  • 27 Henson, S.M., Cushing, J.M., 1997. The effect of periodic habitat fluctuations on a nonlinear insect population model. J. Math. Biol. 36, 201–226.

    Article  MATH  MathSciNet  Google Scholar 

  • 28 Jillson, D., 1980. Insect populations respond to fluctuating environments. Nature 288, 699– 700.

    Article  Google Scholar 

  • 29 Kocic, V.L., 2005. A note on nonautonomous Beverton–Holt model. J. Diff. Equ. Appl. 11(4–5), 415–422.

    Article  MATH  MathSciNet  Google Scholar 

  • 30 Kocic, V.L., Ladas, G., 1993. Global behavior of nonlinear difference equations of higher order with applications. In: Mathematics and its Applications, Vol. 256. Kluwer Academic, Dordrecht, The Netherlands.

    MATH  Google Scholar 

  • 31 Kon, R., 2004. A note on attenuant cycles of population models with periodic carrying capacity. J. Diff. Equ. Appl. 10(8), 791–793.

    MATH  MathSciNet  Google Scholar 

  • 32 Kon, R., in press. Attenuant cycles of population models with periodic carrying capacity. J. Diff. Equ. Appl.

  • 33 Kot, M., Schaffer, W.M., 1984. The effects of seasonality on discrete models of population growth. Theor. Popul. Biol. 26, 340–360.

    Article  MATH  MathSciNet  Google Scholar 

  • 34 Li, J., 1992. Periodic solutions of population models in a periodically fluctuating environment. Math. Biosci. 110, 17–25.

    Article  MATH  MathSciNet  Google Scholar 

  • 37 May, R.M., 1974a. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, NJ.

    Google Scholar 

  • 38 May, R.M., 1974b. Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos. Science 186, 645–647.

    Article  Google Scholar 

  • 36 May, R.M., 1977. Simple mathematical models with very complicated dynamics. Nature 261, 459–469.

    Article  Google Scholar 

  • 35 May, R.M., Oster, G.F., 1976. Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573–579.

    Article  Google Scholar 

  • 39 Moran, P.A.P., 1950. Some remarks on animal population dynamics. Biometrics 6, 250–258.

    Article  Google Scholar 

  • 40 Nicholson, A.J., 1954. Compensatory reactions of populations to stresses, and their evolutionary significance. Aust. J. Zool. 2, 1–65.

    Article  Google Scholar 

  • 41 Nisbet, R.M., Gurney, W.S.C., 1982. Modelling Fluctuating Populations, Wiley, New York.

    MATH  Google Scholar 

  • 42 Nobile, A., Ricciardi, L.M., Sacerdote, L., 1982. On Gompertz growth model and related difference equations. Biol. Cybern. 42, 221–229.

    MATH  Google Scholar 

  • 43 Pennycuick, C.J., Compton, R.M., Beckingham, L., 1968. A computer model for simulation the growth of a population, or of two interacting populations. J. Theor. Biol. 18, 316– 329.

    Article  Google Scholar 

  • 44 Ricker, W.E., 1954. Stock and recruitment. J. Fish. Res. Bd. Can. 11, 559–623.

    Google Scholar 

  • 45 Rodriguez, D.J., 1988. Models of growth with density regulation in more than one life stage. Theor. Popul. Biol. 34, 93–117.

    Article  MATH  Google Scholar 

  • 46 Rosenblat, S., 1980. Population models in a periodically fluctuating environment. J. Math. Biol. 9, 23–36.

    Article  MATH  MathSciNet  Google Scholar 

  • 47 Rosenkranz, G., 1983. On global stability of discrete population models. Math. Biosci. 64, 227– 231.

    Article  MATH  MathSciNet  Google Scholar 

  • 48 Selgrade, J.F., Roberds, H.D., 2001. On the structure of attractors for discrete, periodically forced systems with applications to population models. Physica D 158, 69–82.

    Article  MATH  MathSciNet  Google Scholar 

  • 49 Smith, J.M., 1968. Mathematical ideas in biology. Cambridge University Press, Cambridge, England.

    Google Scholar 

  • 50 Smith, J.M., 1974. Models in Ecology. Cambridge University Press, Cambridge, UK.

    MATH  Google Scholar 

  • 51 Utida, S., 1957. Population fluctuation,an experimental and theoretical approach. Cold Spring Harbor Symp. Quant. Biol. 22, 139–151.

    Google Scholar 

  • 52 Yakubu, A.-A. 2005. Periodically forced nonlinear difference equations with delay. In: Allen, L., Aulbach, B., Elaydi, S., Sacker, R. (Eds.), Difference equations and discrete dynamical systems, Proceedings of the 9th International Conference, University of Southern California, California. World Sci. Publ., Hackensack, NJ, pp. 217–231.

    Google Scholar 

  • 53 Yodzis, P., 1989. Introduction to Theoretical Ecology. Harper and Row, New York.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA

    John E. Franke

  2. Department of Mathematics, Howard University, Washington, DC, 20059, USA

    Abdul-Aziz Yakubu

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  1. John E. Franke
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  2. Abdul-Aziz Yakubu
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Correspondence to Abdul-Aziz Yakubu.

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Franke, J.E., Yakubu, AA. Signature Function for Predicting Resonant and Attenuant Population 2-cycles. Bull. Math. Biol. 68, 2069–2104 (2006). https://doi.org/10.1007/s11538-006-9086-8

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