Systems of nonlinear delay integral equations modelling population growth in a periodic environment

Antonio Cañada; Abderrahim Zertiti

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 4, page 633-644
  • ISSN: 0010-2628

Abstract

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In this paper we study the existence and uniqueness of positive and periodic solutions of nonlinear delay integral systems of the type x ( t ) = ∫ t - τ 1 t f ( s , x ( s ) , y ( s ) ) d s y ( t ) = ∫ t - τ 2 t g ( s , x ( s ) , y ( s ) ) d s which model population growth in a periodic environment when there is an interaction between two species. For the proofs, we develop an adequate method of sub-supersolutions which provides, in some cases, an iterative scheme converging to the solution.

How to cite

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Cañada, Antonio, and Zertiti, Abderrahim. "Systems of nonlinear delay integral equations modelling population growth in a periodic environment." Commentationes Mathematicae Universitatis Carolinae 35.4 (1994): 633-644. <http://eudml.org/doc/247604>.

@article{Cañada1994,
abstract = {In this paper we study the existence and uniqueness of positive and periodic solutions of nonlinear delay integral systems of the type \[ x(t) = \int \_\{t-\tau \_1\}^t f(s,x(s),y(s))\,ds \]\[ y(t) = \int \_\{t-\tau \_2\}^t g(s,x(s),y(s))\,ds \] which model population growth in a periodic environment when there is an interaction between two species. For the proofs, we develop an adequate method of sub-supersolutions which provides, in some cases, an iterative scheme converging to the solution.},
author = {Cañada, Antonio, Zertiti, Abderrahim},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonlinear integral equations; monotone methods; population dynamics; positive solutions; existence; uniqueness; systems of nonlinear Volterra integral equations; population growth; positive solution},
language = {eng},
number = {4},
pages = {633-644},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Systems of nonlinear delay integral equations modelling population growth in a periodic environment},
url = {http://eudml.org/doc/247604},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Cañada, Antonio
AU - Zertiti, Abderrahim
TI - Systems of nonlinear delay integral equations modelling population growth in a periodic environment
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 4
SP - 633
EP - 644
AB - In this paper we study the existence and uniqueness of positive and periodic solutions of nonlinear delay integral systems of the type \[ x(t) = \int _{t-\tau _1}^t f(s,x(s),y(s))\,ds \]\[ y(t) = \int _{t-\tau _2}^t g(s,x(s),y(s))\,ds \] which model population growth in a periodic environment when there is an interaction between two species. For the proofs, we develop an adequate method of sub-supersolutions which provides, in some cases, an iterative scheme converging to the solution.
LA - eng
KW - nonlinear integral equations; monotone methods; population dynamics; positive solutions; existence; uniqueness; systems of nonlinear Volterra integral equations; population growth; positive solution
UR - http://eudml.org/doc/247604
ER -

References

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  4. Ca nada A., Zertiti A., Topological methods in the study of positive solutions for some nonlinear delay integral equations, to appear in J. Nonlinear Analysis. MR1305767
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  6. Guo D., Lakshmikantham V., Positive solutions of integral equations arising in infectious diseases, J. Math. Anal. Appl. 134 (1988), 1-8. (1988) MR0958849
  7. Krasnoselskii M.A., Positive solutions of operator equations, P. Noordhoff, Groningen, The Netherlands, 1964. MR0181881
  8. Nussbaum R.D., A periodicity threshold theorem for some nonlinear integral equations, Siam J. Math. Anal. 9 (1978), 356-376. (1978) Zbl0385.45007MR0477924
  9. Smith H.L., On periodic solutions of a delay integral equations modelling epidemics, J. Math. Biology 4 (1977), 69-80. (1977) MR0504059
  10. Torrejon R., A note on a nonlinear integral equation from the theory of epidemics, J. Nonl. Anal. 14 (1990), 483-488. (1990) Zbl0695.45008MR1044076

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