Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients

Juan C. Jara; Felipe Rivero

doi:10.3934/dcds.2014.34.4127

Article Contents

2014, Volume 34, Issue 10: 4127-4137. Doi: 10.3934/dcds.2014.34.4127

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Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients

Juan C. Jara^1, and
Felipe Rivero^2,

1.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tarfia s/n, 41012-Seville
2.
Departamento de Matemática y Mecánica, I.I.M.A.S - U.N.A.M., Apdo. Postal 20-726, 01000 México D. F.

Received: September 2013

Revised: October 2013

Published: October 2014

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Abstract

In this work we study the asymptotic behaviour of the following prey-predator system \begin{equation*} \left\{ \begin{split} &A'=\alpha f(t)A-\beta g(t)A^2-\gamma AP\\ &P'=\delta h(t)P-\lambda m(t)P^2+\mu AP, \end{split} \right. \end{equation*} where functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are not necessarily bounded above. We also prove the existence of the pullback attractor and the permanence of solutions for any positive initial data and initial time, making a previous study of a logistic equation with unbounded terms, where one of them can be negative for a bounded interval of time. The analysis of a non-autonomous logistic equation with unbounded coefficients is also needed to ensure the permanence of the model.

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Mathematics Subject Classification: 92D15, 35B40, 37N25, 92D25, 34E99.

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