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Competition systems with periodic coefficients: A geometric approach

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The classical two-species competition system is modified to include coefficients which are time-periodic with the same period. We show first that all (nonnegative) solutions converge to a periodic one, having the same period, thus excluding subharmonics. The global structure of the set of all periodic solutions is then investigated. This is accomplished by developing a geometric theory of the discrete dynamical system defined by the iterates of the period map T. It turns out, in particular, that periodic solutions appear which have no counterpart in the corresponding time-averaged system: thus oscillations in the environment may cause the two species to coexist in an oscillatory regime even if the corresponding averaged system would force either of the two species to extinction.

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  1. Bardi, M.: Predator-prey models in periodic environments. University of Padua: Preprint 1980

  2. Cushing, J. M.: Stable positive periodic solutions of the time-dependent logistic equation under possible hereditary influences. J. Math. Anal. Appl. 60, 747–754 (1977)

    Google Scholar 

  3. Cushing, J. M.: Periodic time-dependent predator-prey systems. SIAM J. Appl. Math. 32, 82–95 (1977)

    Google Scholar 

  4. Cushing, J. M.: Two species competition in a periodic environment. J. Math. Biol. 10, 385–400 (1980)

    Google Scholar 

  5. Koch, A. L.: Coexistence resulting from an alternation of density dependent and density independent growth, J. Theor. Biol. 44, 373–386 (1974)

    Google Scholar 

  6. Krasnosel'skii, M. A.: The operator of translation along the trajectories of differential equations. Providence, RI: AMS 1968

    Google Scholar 

  7. de Mottoni, P., Schiaffino, A.: On logistic equations with time periodic coefficients. Roma: Pubbl. IAC n. 192, 1979

    Google Scholar 

  8. Pliss, V. A.: Integral'nye množestva periodičeskih sistem differencial'nyh uravnenii. Moskva: Nauka 1977 (Russian)

    Google Scholar 

  9. Rosenblat, S.: Population models in a periodically fluctuating environment. J. Math. Biol. 9, 23–36 (1980)

    Google Scholar 

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de Mottoni, P., Schiaffino, A. Competition systems with periodic coefficients: A geometric approach. J. Math. Biology 11, 319–335 (1981).

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