\documentclass[reqno]{amsart}

\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. 1--8.\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}}
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\begin{document}

\title[\hfilneg EJDE/Conf/12 \hfil Asymptotic behavior of discrete solutions]
{Asymptotic behavior of discrete solutions to impulsive
logistic equations}

\author[H. Ak\c{c}a, E. A. Al-Zahrani, V.  Covachev\hfil EJDE/Conf/12 \hfilneg]
{Haydar Ak\c{c}a, Eada Ahmed Al-Zahrani, Val\'{e}ry Covachev}  % in alphabetical order

\address{Haydar Ak\c{c}a \hfill\break
Department of Mathematical Science, King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia}
\email{akca@kfupm.edu.sa}

\address{Eada Ahmed Al-Zahrani \hfill\break
Department of Mathematics, Science College for Girls, Dammam, Saudi Arabia}
\email{eada00@hotmail.com}

\address{Val\'{e}ry Covachev \hfill\break
Department of Mathematics \& Statistics, College of Science,
Sultan Qaboos University,
Muscat 123, 
Sultanate of Oman \hfill\break
Permanent address:
Institute of Mathematics,
Bulgarian Academy of Sciences,
Sofia 1113, Bulgaria}
\email{vcovachev@hotmail.com, vcovachev@yahoo.com, valery@squ.edu.om}

\date{}
\thanks{Published April 20, 2005.}
\thanks{Partially supported by grant SABIC/FAST TRACK/2002-7 from KFUPM}
\subjclass[2000]{34A37}
\keywords{Impulsive differential equation; logistic equation; discrete analogue}

\begin{abstract}
We study the stability characteristics of a time-dependent system of
impulsive logistic equations by using discrete modelling.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{thr}{Theorem}
%\allowdisplaybreaks

\section{Introduction}
One of the most used models for  a single
species dynamics has been derived by many researchers in the form of
a differential equation
\begin{equation}
\dot{x}(t) = x(t)f(t, x(t))-g(t,x(t)), \label{1}
\end{equation}
where the solution $  x(t)  $ is  the size or biomass of the
resource population at time $  t>0  $, the function $ f(t, x(t)) $
characterizes the population change at time $ t $,
and the function $  g(t, x(t))  $ defines the continuous effects ---
influences of external factors. When  $g(t, x(t)) = 0$, then
the model is called classical logistic equation, which
represents a population isolated from external factors.

As an illustrative example, a special case of \eqref{1},  consider the
well known elementary nonlinear ordinary differential equation
\begin{equation}
\frac{dy(t)}{dt} =  y(t)  [ 1- y(t) ], \quad t>0\,. \label{2}
\end{equation}
The autonomous system  has two equilibrium states: $ y(t)=0 $  and
$ y(t)=1 $. It is not  difficult to see that the trivial solution
is unstable while the positive equilibrium of \eqref{2}  is asymptotically
stable and all solutions of \eqref{2}  satisfy $y(0) > 0 \Rightarrow y(t)
>0  $ for  $  t>0  $ and $  y(t) \to1  $ as $ t \to
\infty  $  and the convergence is monotonic. One of the discrete
versions of \eqref{2} is given by the quadratic logistic equation (an
Euler-type scheme)
\begin{equation}
y(n+1) = y(n) [1+h-hy(n) ], \quad h>0,\;  n \in\mathbb{
Z}_{0}^{+}. \label{3}
\end{equation}
The trivial equilibrium of \eqref{3} is unstable for $ h>0$. The
positive equilibrium of \eqref{3} is asymptotically stable for $0<
h \leq 2$, for $h > 2$ it becomes unstable, and oscillatory and
chaotic behavior of solutions of \eqref{3} is possible. See more
details in \cite{11,13,14} and references therein. Equation
\eqref{3} can be reformulated so as to obtain the following
popular logistic equation:
\begin{equation} \label{4}
\begin{gathered}
x(n+1) = rx(n) [1-x(n)],\\
  r =1+h,\quad  x(n) = \frac{h}{1+h} y(n)\,.
\end{gathered}
\end{equation}
It has infinitely many, a one-parameter family of nonzero
equilibria for positive $r$. The dynamical characteristics of
\eqref{4} are well known in the literature on chaotic systems
\cite{8,14}.

Let us consider a second order Runge-Kutta algorithm for
\eqref{2};   such an algorithm leads to a discrete-time system
defined by the following expressions:
\begin{equation} \label{5}
\begin{gathered}
x(n+1) = x(n) + \frac{1}{2} (k_1 + k_2 ),\\
k_1 = hx(n) (1-x(n) ),  \\
k_2 = h( x(n) + k_1 ) ( 1- x(n) - k_1),
\end{gathered}
\end{equation}
where  $  n \in\mathbb{Z}_{0}^{+}$ and $  h>0  $ is a fixed
constant. An algebraic simplification of \eqref{5} leads to
\begin{equation} \label{6}
\begin{aligned}
x(n+1)& = \Big(1+ \frac{h(2+h)}{2} \Big) x(n) - \frac{h(2+3h
+h^2)}{2} x^{2}(n) \\
&\quad +\Bigl( \bigl(1+h \bigr) h^{2} \Bigr) x^{3}(n) - \frac{h^{3}}{2}
x^{4}(n), \quad n \in \mathbb{Z}_0^{+}.
\end{aligned}
\end{equation}
The equilibria of  \eqref{6} are given by $x_i^{*},\;i = 1,2,3,4$,  where
\begin{gather*}
x_1^{*} = 0, \quad x_2^{*} = 1, \\
x_3^{*} = \frac{1}{2h} \big( 2+h-\sqrt{h^{2} -4}\big),\quad
x_4^{*} = \frac{1}{2h} \big(2+h+\sqrt{h^{2}-4}\big).
\end{gather*}
Thus it follows that for $  h>2$, \eqref{6}
has four equilibrium points while its mother version \eqref{2}
has only two equilibria.  Also for  $  h>2,\; x_1^{*} $  and  $
x_2^{*} $ are  unstable while $ x_3^{*} $ and  $ x_4^{*}  $ are
stable.  The equilibria   $ x_3^{*}   $ and    $ x_4^{*} $ are
born through the process of discretization of \eqref{2} and
these are parasitic, spurious or ghost solutions of the system.

Higher order Runge-Kutta numerical algorithms can give rise to
more parasitic solutions including periodic, quasiperiodic and
chaotic behaviour. For more details see \cite{7,9} and
\cite{15,16,17}.

\section{Main result}
Consider the non-autonomous
logistic equation
\begin{gather}
\frac{dx}{dt}  =  r(t)x(t) \big(1 - \frac{x(t)}{K(t)}\big), \quad
t > 0,\; t \neq \tau _k, \label{7}\\
\Delta x(t)   =  I_k (x(t)), \quad t = \tau_k,\; k = 1,2, \dots, \label{8}
\end{gather}
in which $r(t)$ is nonnegative  and $K(t)$ is a strictly positive
continuous function, and $I_k$ are bounded operators. In the real
evolutionary processes of the population, the perturbation or the
influence from outside occurs ``instantly'' as impulses, and not
continuously. The duration of these perturbations is negligible
compared to the duration of the whole process, for more details about
the theory of impulsive differential equations and applications
see \cite{10,18} and references therein. Also impulsive
perturbations (harvest, taking out, hunting, fishing, etc.) are
more practical and realistic compared to any kind of
continuous harvest. For instance, a fisherman can not fish 24
hours a day and furthermore, the seasons also determine the
fishing period. Similar considerations are applicable for hunting
and taking away a huge part of any biomass.

In system \eqref{7}, \eqref{8} the instants of impulse effect $\tau_k$,
$k=1,2,\dots$, form a strictly increasing sequence such that
$\lim_{k\to\infty}\tau_k=+\infty$. By a solution of
system \eqref{7}, \eqref{8} we mean a piecewise continuous function on
$[0,+\infty)$ which satisfies the equation \eqref{7} for $t>0$, $t\neq
\tau_k$, with discontinuities of the first kind at $\tau_k$, $k=1,2,\dots$, at
 which it is continuous from the left and satisfies the impulsive conditions
$$\Delta x(\tau_k)\equiv x(\tau_k+0)-x(\tau_k)=I_k(x(\tau_k)).$$

The logistic equation
\eqref{7} has been intensively studied by various researchers
\cite{1,2,4} and \cite{5,6}, considering existence  of the
solutions, asymptotic properties of the solutions, sufficient
conditions for the oscillation of the solutions and so on. In this
paper we will use a suitable differential equation with piecewise
constant argument to approximate the solution of the system
\eqref{7}, \eqref{8} and introduce a sufficient condition for the
existence of the solution of the impulsive systems. The
approximation is given by
\begin{equation} \label{9}
\begin{gathered}
\frac{dx}{dt}  =  x(t)
\Big(r\big(\big[\frac{t}{h}\big]h\big) -
\frac{r([\frac{t}{h}]h)x(t)}{K([
\frac{t}{h}]h)}\Big),\\
 t\in[nh,(n+1)h),\; n\in\mathbb{Z}_0^+,\;n \neq\big[\frac{\tau_k}h\big], \\
x\Big(\bigl(\big[\frac{\tau _k}{h}\big]+1\bigr)h\Big)  =
x\big(\big[\frac{\tau _k}{h}\big]h\big)+
I_k x\big(\big[\frac{\tau _k}{h}\big]h\big), \quad k = 1,2, \dots,
\end{gathered}
\end{equation}
where $h > 0$ denotes
a uniform discretization step size and $[\mu]$ denotes
the integer part of $\mu \in \mathbb{R}$. Then the system \eqref{7}, \eqref{8}
becomes
\begin{equation}  \label{10}
\begin{gathered}
\frac{dx}{dt}  =  r(n) x(t) - \frac{r(n)}{K(n)}
x^2(t),\quad t\in[nh,(n+1)h),\;n\neq m_k,\\
x(m_k+1)  = x(m_k)+ I_k (x(m_k)), \quad k = 1, 2, \dots,
\end{gathered}
\end{equation}
where $\left[ \frac{t}{h}\right] = n, \left[
\frac{\tau _k}{h}\right] = m_k$, and we use the notation
$f(n) = f(nh)$.

An integration of the differential equation in \eqref{10} over $[nh, t)$,
where $t < (n+1)h$,
leads to
$$
 \frac{1}{x(t)} e^{r(n)t} - \frac{1}{x(n)}
e^{r(n)nh} = \frac{e^{r(n)t}}{K(n)} -
\frac{e^{r(n)nh}}{K(n)}, \quad \tau _k\not\in[nh,t),
$$
and by allowing $t\to(n+1)h$, we obtain after some
simplification
\begin{equation} \label{11}
\begin{gathered}
x(n+1) = \frac{e^{r(n)h}x(n)}{1 +
\big(\frac{e^{r(n)h}-1}{K(n)}\big)x(n)}, \quad n\neq m_k,  \\
x (m_k+1) = x(m_k)+ I_k (x(m_k)), \quad k = 1, 2, \dots 
\end{gathered}
\end{equation}

\begin{thr} \label{thr1}
Let the following conditions hold:
\begin{gather*}
r(n)\ge 0,\;n\in \mathbb{Z}_0^+,\quad
 R= \sup_{n\in \mathbb{Z}_0^+} r(n) < \infty ,\quad
0 < K_* \leq \inf_{n\in\mathbb{Z}_0^+} K(n),\\
\sup_{n\in \mathbb{Z}_0^+} K(n) < \infty,\quad I_k (x(m_k)) = cx(m_k),
\end{gather*}
where $c>0$. Then for $h>0$ satisfying the inequality
$h\le \ln(1+c)/R$
a solution $x(n)$ of $\eqref{11}$ corresponding to
$x(0) > 0$ satisfies the
inequality
\begin{equation}
\frac{1}{x(n)} \leq \frac{1}{x(0)} \exp\
\Big(- \sum^{n-1}_{i=0} r(i)h\Big)
+ \sum^n_{j=1} \frac{1-e^{-r(n-j)h}}{K(n-j)} 
\exp \Big(-  \sum^{j-1}_{\ell=1} r(n-\ell)h\Big).
\label{12}
\end{equation}
\end{thr}

\begin{proof}
 Since $x(n) > 0$ for all $n\in \mathbb{Z}^+_0$, we let
$ y(n) ={\displaystyle \frac{1}{x(n)}}$ in \eqref{11}, and obtain
\begin{equation} \label{13}
\begin{gathered}
y(n+1) = e^{-r(n)h}y(n) + \frac{1-e^{-r(n)h}}{K(n)},
\quad n\in \mathbb{Z}^+_0,\; n\neq m_k,  \\
y(m_k+1) = \frac{1}{1+c}y(m_k), \quad k = 1,2, \dots
\end{gathered}
\end{equation}
We will show by induction that \eqref{13} leads to
\begin{equation}
\begin{aligned}
y(n) &\leq y(0) \exp\Big(- \sum^{n-1}_{i=0} r(i)h\Big)\\
&\quad + \sum^{n-1}_{j=0} \frac{1}{K(j)}
\Big\{ \exp\Big(- \sum^{n-1}_{\ell=j+1}r(\ell)h\Big) -
\exp\Big(-\! \sum^{n-1}_{\ell=j} r(\ell)h\Big) \Big\}. \label{14}
\end{aligned}
\end{equation}
In fact, for $n=0$ \eqref{14} is obviously true. Suppose that it is true
for some $n\ne m_k$. Then from \eqref{13} we find
\begin{align*}
&y(n+1)\\
& = e^{-r(n)h}y(n) + \frac{1-e^{-r(n)h}}{K(n)} \\
&\leq  e^{-r(n)h}y(0) \exp\Big(- \sum^{n-1}_{i=0} r(i)h\Big)\\
&\quad + e^{-r(n)h} \sum^{n-1}_{j=0} \frac{1}{K(j)}
\Big\{ \exp\Big(- \sum^{n-1}_{\ell=j+1}r(\ell)h\Big) -
\exp\Big(- \sum^{n-1}_{\ell=j} r(\ell)h\Big)\Big\} +
\frac{1-e^{-r(n)h}}{K(n)} \\
&=y(0) \exp\Big(- \sum^{n}_{i=0} r(i)h\Big) +
\sum^{n-1}_{j=0} \frac{1}{K(j)}
\Big\{\exp\Big(- \sum^{n}_{\ell=j+1}r(\ell)h\Big) -
\exp\Big(- \sum^{n}_{\ell=j} r(\ell)h\Big)\Big\}\\
&\quad +\frac{1}{K(n)} \Big\{
\exp\Big(- \sum^{n}_{\ell=n+1}r(\ell)h\Big) -
\exp\Big(- \sum^{n}_{\ell=n} r(\ell)h\Big)\Big\}\\
&= y(0) \exp\Big(- \sum^{n}_{i=0} r(i)h\Big) +
\sum^{n}_{j=0} \frac{1}{K(j)}
\Big\{\exp\Big(- \sum^{n}_{\ell=j+1}r(\ell)h\Big) -
\exp\Big(- \sum^{n}_{\ell=j} r(\ell)h\Big)\Big\}.
\end{align*}
Next  suppose that inequality \eqref{14} is true for $n=m_k$. Then,
using the inequality $ \exp\big(r(m_k)h\big)\le 1+c$ which follows
from our assumptions, we obtain
\begin{align*}
&y(m_k+1)\\
&=\frac1{1+c}y(m_k)\\
&\le e^{-r(m_k)h}y(m_k) \\
&\le y(0) \exp\Big(\!-\! \sum^{m_k}_{i=0} r(i)h\Big) +
\sum^{m_k-1}_{j=0} \frac{1}{K(j)}
\Big\{\exp\Big(\!-\! \sum^{m_k}_{\ell=j+1}r(\ell)h\Big) -
\exp\Big(\!-\! \sum^{m_k}_{\ell=j} r(\ell)h\Big)\Big\}\\
&\le y(0) \exp\Big(\!-\! \sum^{m_k}_{i=0} r(i)h\Big) +
\sum^{m_k}_{j=0} \frac{1}{K(j)}
\Big\{\exp\Big(\!-\! \sum^{m_k}_{\ell=j+1}r(\ell)h\Big) -
\exp\Big(\!-\! \sum^{m_k}_{\ell=j} r(\ell)h\Big)\Big\}.
\end{align*}
Thus \eqref{14} is proved.
By changing the summation variables in \eqref{14}, we get
\begin{equation}
y(n) \leq y(0) \exp\Big(- \sum^{n-1}_{i=0}r(i)h\Big)
+ \sum^n_{m=1} \frac{1-e^{-r(n-m)h}}{K(n-m)}
\exp\Big(- \sum^{m-1}_{\ell=1}r(n-\ell)h\Big). \label{15}
\end{equation}
Since $y(n) > 0$ for all $n\in \mathbb{Z}_0^+$, we can substitute
$ x(n) = \frac{1}{y(n)},\; n\in \mathbb{Z}_0^+$, and this completes
the proof.
\end{proof}

Now we will study the asymptotic behavior of the solutions of \eqref{11} where
$r(\cdot)$ and $K(\cdot)$ are time dependent.

\begin{thr}\label{thr2}
Let all assumptions of Theorem \ref{thr1} hold and suppose further
that there exists a number $\hat{r} > 0$ such that
\begin{equation}
\lim_{m\to\infty} \frac{1}{m} \Big\{\sum^m_{j=1}
r(n-j)\Big\}
= \hat{r},\; m\in \mathbb{Z}^+, \mbox{ uniformly on } n\in\mathbb{Z}.
\label{16}
\end{equation}
Then the solution $x(n)$ of the system $\eqref{11}$ tends to $x^*(n)$
as $n\to\infty$, where $x^*(n)$ is given by
$$
x^*(n) = \Big[\sum^\infty_{j=1}  \frac{1 -
e^{-r(n-j)h}}{K(n-j)}
\exp\Big(- \sum^{j-1}_{\ell=1} r(n-\ell)h\Big)\Big]^{-1}
$$
in the sense that $x(n)-x^*(n) \to0$ as $n\to
\infty$.
\end{thr}

\begin{proof} Since $x(n)$ is positive, we can use
$ y(n) = 1/x(n)$, $n\in \mathbb{Z}^+_0$, and from
Theorem \ref{thr1} we have
$$
y(n) \leq y(0) \exp\Big(- \sum^{n-1}_{i=0} r(i)h\Big) +
\sum^n_{j=1} \frac{1-e^{-r(n-j)h}}{K(n-j)}
\exp\Big(- \sum^{j-1}_{\ell=1} r(n-\ell)h\Big),
$$
then
\begin{align*}
& y(n) - \sum^\infty_{j=1} 
\frac{1-e^{-r(n-j)h}}{K(n-j)}
\exp\Big(- \sum^{j-1}_{\ell=1} r(n-\ell)h\Big) \\
&\le  y(0) \exp\Big(- \sum^{n-1}_{i=0} r(i)h\Big) -
\sum^\infty_{j=n+1} \frac{1- e^{-r(n-j)h}}{K(n-j)}
\exp\Big(- \sum^{j-1}_{\ell=1} r(n-\ell)h\Big)
\end{align*}
and hence
\begin{equation} \label{17}
\begin{aligned}
&\Big|y(n) - \sum^\infty_{j=1} 
\frac{1-e^{-r(n-j)h}}{K(n-j)} 
\exp\Big(- \sum^{j-1}_{\ell=1} r(n-\ell)h\Big)\Big|\\
&\leq y(0) \exp\Big(- \sum^{n-1}_{i=0}r(i)h\Big)
+ \sum^\infty_{j=n+1} \frac{1-e^{-r(n-j)h}}{K(n-j)}
\exp\Big(- \sum^{j-1}_{\ell=1} r(n-\ell)h\Big).
\end{aligned}
\end{equation}
Let us choose a number $\xi$ satisfying $0 < \xi <
\hat{r}$.
{From} the assumption \eqref{16}, there corresponds a positive
integer $N=N(\xi)$ such that
$$
n(\hat{r}-\xi)h \leq \sum^{n-1}_{i=0} r(i)h \leq n (\hat{r}
+\xi) h \quad \forall\; n\geq N.
$$
By substituting into \eqref{17}, for $n\ge N$ we obtain
\begin{align*}
&\Big| y(n) - \sum^\infty_{j=1} 
\frac{1-e^{-r(n-j)h}}{K(n-j)} 
\exp\Big(-\sum^{j-1}_{\ell=1}r(n-\ell)h\Big)\Big| \\
&\leq   y(0) e^{-n(\hat{r}-\xi)h} +
\frac{1}{K_*} \sum^\infty_{j=n+1} e^{-(j-1)(\hat{r}-\xi)h}\\
&=  y(0) e^{-n(\hat{r}-\xi)h} +
\frac{1}{K_*} \frac{e^{-n(\hat{r}-\xi)h}}{1-e^{-(\hat{r}-\xi)h}}\;
\to0\quad  \mbox{ as } n\to\infty.
\end{align*}
Let us denote
$$
y^*(n) = \sum^\infty_{j=1}
\frac{1-e^{-r(n-j)h}}{K(n-j)}
\exp\Big(- \sum^{j-1}_{\ell=1}r(n-\ell)h\Big)\,.
$$
Then
$y(n)-y^*(n) \to0$ as $n\to\infty$, $y^*(n) > 0$ for all
$n\in \mathbb{Z}$.  In the following, we will show that $y^*(n) < \infty$,
{\em i.e.},
the series $y^*(n)$ is convergent with respect to $n\in \mathbb{Z}$.
As above, for the number $\xi$ there corresponds a
positive integer $N= N(\xi)$ such that
\begin{equation}
m(\hat{r}-\xi)h\leq \sum^m_{j=1} r(n-j)h\leq
m(\hat{r}+\xi)h \mbox{ for } m \geq N. \label{18}
\end{equation}
Now, let us consider the series $ \sum^\infty_{j=1}
\exp\Big(-\sum^{j-1}_{\ell=1} r(n-\ell)h\Big)$ and
rewrite it as
$$
\sum^\infty_{j=1} \exp\Big(- \sum^{j-1}_{\ell=1} r(n-\ell)h\Big)
=\sum^N_{j=1} \exp\Big(- \sum^{j-1}_{\ell=1}r(n-\ell)h\Big)
+\sum^\infty_{j=N+1} \exp\Big(\sum^{j-1}_{\ell=1}r(n-\ell)h\Big).
$$
We note that the integer $N$ is positive and finite and
the first sequence in the above equality is bounded and non-negative
for all $n\in \mathbb{Z}$.  Thus there exists a finite positive real number
$A$ for which $ \sum^N_{j=1} \exp\Big(-\sum^{j-1}_{\ell=1}r(n-\ell)h\Big)
\leq A$ for all $n\in \mathbb{Z}$.

Furthermore, we have that
$$
\sum^\infty_{j=N+1} \exp\Big(-\sum^{j-1}_{\ell=1}r(n-\ell)h\Big) \leq
\frac{e^{-N(\hat{r}-\xi)h}}{1-e^{-(\hat{r}-\xi)h}}.
$$
Thus we have
$$
y^*(n) \leq \frac{1}{K_*}\Big\{A + \frac{e^{-N(\hat{r}-\xi)h}}{1
- e^{-(\hat{r}-\xi)h}}\Big\}.
$$
As above, using the right-hand side in \eqref{18} and the condition
$\sup_{n\in\mathbb{Z}_0^+}K(n) <\infty$, we find that
$\liminf_{n\to\infty}y^*(n)>0$.
Since $y(n)>0$ for all $n$ and $y(n)-y^*(n)\to0$ as $n\to\infty$, we have
$\liminf_{n\to\infty}y(n)>0$ and $\sup_{n\in\mathbb{Z}_0^+}y(n)<\infty$.
Thus $x(n)$ satisfies
$$
\lim_{n\to\infty} (x(n) - x^*(n) )=
\lim_{n\to\infty} \big(\frac{1}{y(n)} -
\frac{1}{y^*(n)}\big) = 0
$$
and the proof is complete. \end{proof}

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\end{document}