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\AtBeginDocument{{\noindent\small
2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.\newline
{\em Electronic Journal of Differential Equations},
Conference 12, 2005, pp. 21--27.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{21}
\begin{document}
\title[\hfilneg EJDE/Conf/12 \hfil Oscillation and asymptotic stability]
{Oscillation and asymptotic stability of a delay differential
equation with Richard's nonlinearity}
\author[L. Berezansky, L. Idels \hfil EJDE/Conf/12 \hfilneg]
{Leonid Berezansky, Lev Idels} % in alphabetical order
\address{Leonid Berezansky\hfill\break
Department of Mathematics \\
Ben-Gurion University of the Negev\\
Beer-Sheva 84105, Israel}
\email{brznsky@cs.bgu.ac.il\; Phone 972-7-6461602\; Fax 972-7-6281340}
\address{Lev Idels \hfill\break
Mathematics Department, Malaspina University-College\\
900 Fifth Street Nanaimo, BC V9R 5S5, Canada}
\email{lidels@shaw.ca\; Phone 250-753-3245 ext. 2429\; Fax 250-740-6482}
\date{}
\thanks{Published April 20, 2005.}
\subjclass[2000]{34K11, 34K20, 34K60}
\keywords{Delay differential equations; Richard's nonlinearity; \hfill\break\indent
oscillation; stability}
\begin{abstract}
We obtain sufficient conditions for oscillation of solutions,
and for asymptotical stability of the positive equilibrium, of
the scalar nonlinear delay differential equation
$$
\frac{dN}{dt} = r(t)N(t)\Big[a-\Big(\sum_{k=1}^m b_k N(g_k(t))\Big)^{\gamma}\Big],
$$
where $ g_k(t)\leq t$.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\section{Introduction}
Consider the following logistic differential equation which is widely used
in Population Dynamics
$$
\frac{dN}{dt}=rN\big( 1-\frac{N}{K}\big).
$$
Here $N(t)$ is the size of a population, $r\geq 0$ is an intrinsic growth
rate, $K$ is a carrying capacity or a saturation level.
A large variety of nonlinear differential equations, besides the one above,
has been developed for models of Mathematical Biology; see for example
\cite{Br,Kot,Baker}.
To model processes in nature and engineering it is frequently
required to know system states from the past. Depending on the phenomena
under study the after-effects represent duration of some hidden processes.
In general, delay differential equations (DDE) exhibit much more
complicated dynamics than ordinary differential equations (ODE) since a
time lag can change a stable equilibrium into an unstable one and make
populations fluctuate, they provide
a richer mathematical framework (compared with ordinary differential
equations) for the analysis
of biosystems dynamics.
Models of Population Dynamics, based on nonlinear
DDE's, have attracted much attention in recent years.
The application of delay equations to biomodelling in many cases
is associated with studies
of dynamic phenomena like oscillations, bifurcations, and chaotic behavior.
Time delays
represent an additional level of complexity that can be incorporated in a more
detailed analysis of a particular system.
The delay logistic equation
\begin{equation}
\label{01}
\frac{dN}{dt}=rN\big( 1-\frac{N_{\tau }}{K}\big)
\end{equation}
appeared in 1948 in Hutchinson's paper \cite{Hut},
where $N_{\tau }=N(t-\tau )$, $\tau >0$.
The autonomous equation (\ref{01}) has been extensively investigated by numerous
authors.
The first paper on the oscillation of a non-autonomous logistic delay
differential equation
was published in \cite{Zhang}. Since this publication, the oscillation of the
logistic DDE
as well as its generalizations were studied by many mathematicians. Some
of these results
can be found in the monographs \cite{GL,Gop,EKZ}.
It is a well-known fact, that the traditional logistic model, in
some cases, produces artificially
complex dynamics. Therefore, it would be reasonable to get away from the
specific logistic
form in studying population dynamics and use more general classes of growth
models.
For example, to drop an unnatural symmetry of the logistic curve,
we consider the modified logistic form by Pella and Tomlinson \cite{TW,PT}
or the Richards' growth equation with delay
\begin{equation}
\label{02}
\frac{dN}{dt}=rN\Big[1-\Big(\frac{N_{\tau }}{K}\Big)^{\gamma }\Big].
\end{equation}
According to \cite{TW}, $0<\gamma <1$ is used for
invertebrate populations (examples of invertebrates are insects, worms,
starfish, sponges, squid, plankton, crustaceans, and mollusks), and
$\gamma\geq 1$ is used for the vertebrate populations
(these include amphibians, birds, fish, mammals, and reptiles).
In \cite{Pon} the authors considered \eqref{02} with several delays.
They obtained conditions for existence of positive solutions
and studied so-called long time average stability.
In this paper we obtain oscillation and local stability results for
non-autonomous \eqref{02} with several delays.
\section{Preliminaries}
Our objective is to study the scalar nonlinear delay differential equation
\begin{equation} \label{1}
\dot{N}(t) = r(t)N(t)\Big[a-\Big(\sum_{k=1}^m b_kN(g_k(t))\Big)^{\gamma}
\Big], \quad t\geq 0
\end{equation}
under the following conditions:
\begin{itemize}
\item[(A1)] $r(t)$ is Lebesgue measurable essentially
bounded on $[0,\infty)$ function, $r(t)\geq 0$.
\item[(A2)] $g_k:[0,\infty)\to \mathbb{R}$ are Lebesgue measurable
functions with $ g_k(t)\leq t$ and $\lim_{t\to \infty}g_k(t)=\infty$,
$k=1,\dots,m$.
\item[(A3)] $a>0$, $b_k>0$, $\gamma>0$.
\end{itemize}
Together with \eqref{1}, we consider for $t_0\geq 0$, the initial-value problem
\begin{gather} \label{2}
\dot{N}(t) = r(t)N(t)\Big[a-\Big(\sum_{k=1}^m b_kN(g_k(t))\Big)^{\gamma}
\Big], \quad t\geq t_0, \\
N(t)=\varphi(t), \quad t0$.
\end{itemize}
\noindent {\bf Definition.} A locally absolutely continuous
function $x:R\to R$ is called {\em a solution of problem}
(\ref{2})--(\ref{3}), if it satisfies (\ref{2}) for almost all $t\in
[t_0,\infty)$ and (\ref{3}) for $t\leq t_0$.
\begin{lemma}[\cite{Pon}] \label{lem1}
Suppose Conditions (A1)--(A4) hold. Then
problem (\ref{2})-(\ref{3}) has a unique positive solution $N(t)$, $t\geq t_0$.
\end{lemma}
\section{Oscillation Criteria}
{\bf Definition.} We say that a function $y(t)$ is {\em non-oscillatory}
about a number $K$ if $y(t)-K$ is eventually positive or eventually negative.
Otherwise $y(t)$ is {\em oscillatory} about $K$.
Note that (\ref{1}) has a positive equilibrium,
\[
N^{\ast}=a^{1/\gamma} / \sum_{k=1}^m b_k.
\]
In this section we study oscillation of solutions of (\ref{1})
about the value $N^{\ast}$.
We will present here some lemmas which will be used in this section.
Consider the linear delay differential equation
\begin{equation} \label{4}
\dot{x}(t)+\sum_{k=1}^l r_k(t)x(h_k(t))=0, \quad t\geq 0,
\end{equation}
and the differential inequalities
\begin{gather}
\label{5}
\dot{x}(t)+\sum_{k=1}^l r_k(t)x(h_k(t))\leq 0, \quad t\geq 0, \\
\label{6}
\dot{x}(t)+\sum_{k=1}^l r_k(t)x(h_k(t))\geq 0, \quad t\geq 0.
\end{gather}
\begin{lemma}[\cite{GL}] \label{lem2}
Let (A1)--(A2) hold for the parameters of \eqref{4}. Then
the following three statements are equivalent:
\begin{enumerate}
\item There exists a non-oscillatory solution of equation \eqref{4}.
\item There exists an eventually positive solution of the inequality
\eqref{5}.
\item There exists an eventually negative solution of the inequality
\eqref{6}.
\end{enumerate}
\end{lemma}
\begin{lemma}[\cite{GL}] \label{lem3}
Let (A1)--(A2) hold for the parameters of \eqref{4}. If
\begin{equation} \label{7}
\liminf_{t\to\infty} \int_{\max_{k}
h_k(t)}^t \sum_{i=1}^l r_i(s)ds > 1/e,
\end{equation}
then all solutions of \eqref{4} are oscillatory.
\end{lemma}
\begin{theorem} \label{thm1}
Suppose (A1)-(A4) hold and
\begin{equation} \label{8}
\int_0^{\infty} r(s)ds =\infty.
\end{equation}
Then for every non-oscillatory solution $N(t)$ of (\ref{1}) we have
\begin{equation} \label{9}
\lim_{t\to\infty} N(t)=N^{\ast}.
\end{equation}
\end{theorem}
\begin{proof} After the substitution
$ %\label{10}
N(t)=N^{\ast}(1+x(t))$,
Equation \eqref{1} reduced to
\begin{equation} \label{11}
\dot{x}(t)=-ar(t)(1+x(t))\Big[\Big(\sum_{k=1}^m B_k(1+x(g_k(t)))\Big)^{\gamma}
-1\Big], \quad t\geq 0,
\end{equation}
where
\begin{equation}
\label{11a}
B_k= b_k /\sum_{i=1}^m b_i.
\end{equation}
Condition (A3) implies $B_k>0$ and $\sum_{k=1}^m B_k=1$.
The zero solution is an equilibrium of
\eqref{11}, which corresponds to the equilibrium $N^{\ast}$ of
\eqref{1}.
By Lemma \ref{lem1} any solution of \eqref{1} is positive. Then for any solution
of (\ref{11}) we have $1+x(t)>0$.
To prove the theorem we have to show that for every non-oscillatory about
zero solution of (\ref{11}) we have
\begin{equation} \label{12}
\lim_{t\to\infty} x(t)=0.
\end{equation}
Suppose $x(t)$ is a non-oscillatory solution of (\ref{11}).
Without loss of generality we can assume
$x(t)>0$, $t\geq 0$. Hence
$$
\Big(\sum_{k=1}^m B_k(1+x(g_k(t)))\Big)^{\gamma}-1\geq
\Big(\sum_{k=1}^m B_k\Big)^{\gamma}-1=0.
$$
Then $\dot{x}(t)\leq 0$ and hence there exists
$\lim_{t\to\infty} x(t)=l$.
Suppose $l>0$. Equality (\ref{11}) implies
\begin{equation}
\label{13}
x(t)=x(0)-a\int_0^t r(s)(1+x(s))\Big[\Big(\sum_{k=1}^m
B_k(1+x(g_k(s)))\Big)^{\gamma}-1\Big]ds.
\end{equation}
If $t\to \infty $ then the right hand side of (\ref{13}) tends to $-\infty$,
the left hand side has a finite limit.
This contradiction proves the theorem.
\end{proof}
\begin{theorem} \label{thm2}
Suppose conditions (A1)--(A4) and \eqref{8} hold,
$\gamma>1$ and there exists $\epsilon>0$ such that all solutions
of the linear differential equation
\begin{equation} \label{14}
\dot{y}(t)=-a\gamma r(t)(1-\epsilon)\sum_{k=1}^m B_k y(g_k(t))
\end{equation}
are oscillatory, were $B_k$ are denoted by \eqref{11a}.
Then all solutions of \eqref{1} are oscillatory about $N^{\ast}$.
\end{theorem}
\begin{proof}
It is sufficient to prove, that all solutions of (\ref{11}) are oscillatory
about zero.
Suppose there exists a non-oscillatory solution $x$ of (\ref{11}).
Without loss of generality we can assume, that $x(t)>0, t\geq 0$.
Theorem \ref{thm1} implies, that for some $t_0>0$ and for $t\geq t_0$ we have
$01$,
\begin{equation} \label{15}
\liminf_{t\to\infty} a\gamma \int_{\max_{ k}
g_k(t)}^t \quad r(s)ds > 1/e.
\end{equation}
Then all solutions of \eqref{1} are oscillatory about $N^{\ast}$.
\end{corollary}
\begin{proof}
Inequality (\ref{15}) implies, that for some $\epsilon>0$,
$$
\liminf_{t\to\infty} a\gamma (1-\epsilon)\int_{\max_{ k}
g_k(t)}^t ~~\sum_{i=1}^m B_i r(s)ds > 1/e.
$$
Lemma \ref{lem3} and Theorem \ref{thm2} imply this corollary.
\end{proof}
\section{Asymptotic Stability}
Consider a general nonlinear delay differential equation
\begin{equation} \label{16}
\dot{x}(t)=f(t,x(t), x(g_1(t)),\dots,x(g_m(t))), \quad t\geq 0,
\end{equation}
with the initial function and the initial value
\begin{equation} \label{17}
x(t)=\varphi(t), \quad t<0, \quad x(0)=x_0,
\end{equation}
under the following conditions:
\begin{itemize}
\item[(B1)] $f(t, u_0, u_1,\dots, u_m)$ satisfies Caratheodory conditions:
Lebesgue measurable in the first argument and continuous in
other arguments, $f(t,0,\dots, 0)=K$
\item[(B2)] $g_k(t)$ are Lebesgue measurable functions,
$$
g_k(t)\leq t, \quad \sup_{t\geq 0}[t-g_k(t)]<\infty;
$$
\item[(B3)] $\varphi :(-\infty,0)\to R $ is a Borel
measurable bounded function.
\end{itemize}
We will assume that the initial-value problem \eqref{16}--\eqref{17}
has a unique global solution $x(t)$, $t\geq 0$.
\noindent{\bf Definition.}
We will say that the equilibrium $K$ of (\ref{16}) is {\em (locally)
stable}, if for any $\epsilon>0 $ there exists
$\delta>0$ such that for every initial conditions $|x(0)|<\delta_0$,
$|\varphi(t)|<\delta_0$, $\delta_0\leq \delta$, for the solution $x(t)$
of (\ref{16})--(\ref{17}) we have $|x(t)-K|<\epsilon$, $t\geq 0$.
If, in addition, $\lim_{t\to\infty}(x(t)-K)=0$, then the equilibrium $K$ of
(\ref{16}) is {\em (locally) asymptotically stable}.
Suppose there exist $M>0$, $\gamma>0$ such that
$$
|x(t)-K|\leq M\exp\{-\gamma t\}(|x(0)|+\sup_{t<0}|\varphi(t)|)
$$
for all $x(0)$ and $\varphi(t)$ such that $|x(0)|+\sup_{t<0}|\varphi(t)|$
is sufficiently small. Then we will say that the equilibrium $K$ of (\ref{16}) is
{\em exponentially stable}.
\begin{lemma}[\cite{Kr}] \label{lem4}
Suppose (A1), (B2), (B3) hold for the linear equation \eqref{4} and
$$
\limsup_{t\to\infty} \sum_{k=1}^l r_k(t)(t-h_k(t))<1.
$$
Then \eqref{4} is exponentially stable.
\end{lemma}
\begin{lemma}[\cite{BC}, \cite{KM}] \label{lem5}
Suppose that (b1)-(b3) hold, and that for sufficiently small $u$ if
$|u_k|\leq u$, $k=0,\dots,m$ then
$$
|f(t,u_0,\dots,u_m)-\sum_{k=0}^m \frac{\partial F}{\partial u_k}(t,K,\dots,K)u_k|
=o(u),
$$
where $\lim_{u\to 0} o(u)/u=0$.
If the linear equation
$$
\dot{y}(t)=\sum_{k=0}^m \frac{\partial F}{\partial u_k}(t,0,\dots,0)\, y(g_k(t))
$$
is exponentially stable, then the equilibrium $K$ of \eqref{16} is locally
asymptotically stable.
\end{lemma}
\begin{theorem} \label{thm3}
Suppose that for equation (\ref{1}) Conditions (A1), (A3), (B2), (B3) hold and
\begin{equation} \label{18}
\limsup_{t\to\infty} a\gamma r(t)\sum_{k=1}^m B_k(t-g_k(t))<1,
\end{equation}
were $B_k$ are denoted by \eqref{11a}.
Then the equilibrium $N^{\ast}$ of \eqref{1} is asymptotically stable.
\end{theorem}
\begin{proof}
The substitution $N(t)=N^{\ast}(1+x(t))$ implies that the
equilibrium $N^{\ast}$ of \eqref{1} is
asymptotically stable if and only if the zero solution of \eqref{11}
is asymptotically stable.
Lemma \ref{lem4} and inequality \eqref{18} imply that the linear equation
$$
\dot{x}(t)=-a\gamma r(t)\sum_{k=1}^m B_kx(g_k(t))
$$
is exponentially stable.
Lemma \ref{lem5} implies now that the zero solution of (\ref{11}) is
asymptotically stable.
\end{proof}
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\end{document}