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Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology

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Abstract

With appropriate assumptions, the following two general first-order nonlinear differential delay equations may be employed to describe some physiological control systems as well as some population growth processes:

$$x'(t) = f\left( {\int_{ - \tau }^{ - \sigma } x (t + s)d\mu (s)} \right) - g(x(t))$$

and

$$x'(t) = x(t)\left[ {f\left( {\int_{ - \tau }^{ - \sigma } x (t + s)d\mu (s)} \right) - g(x(t))} \right].$$

It is assumed thatg(x) is strictly increasing,g(0)=0, and each of these two equations has a unique positive steady state. Sufficient conditions are obtained for this steady state to be a global attractor (with respect to continuous positive initial functions) whenf(x) is monotone or has only one hump. We also establish the global existence of periodic solutions in these equations.

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Authors and Affiliations

  1. Department of Mathematics, Arizona State University, 85287-1804, Tempe, AZ, USA

    Yang Kuang

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  1. Yang Kuang
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Kuang, Y. Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology. Japan J. Indust. Appl. Math. 9, 205 (1992). https://doi.org/10.1007/BF03167566

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  • DOI: https://doi.org/10.1007/BF03167566

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