\input amstex
\documentstyle{amsppt}
\magnification=\magstephalf \hcorrection{1cm} \vcorrection{-6mm}
\nologo
\TagsOnRight \NoBlackBoxes \pageno=41
\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{\hfil Models of population interactions
\hfil\folio}
\def\leftheadline{\folio\hfil Yulin Cao \& Thomas C. Gard
\hfil}
\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
Differential Equations and Computational Simulations III\hfill\break
J. Graef, R. Shivaji, B. Soni, \& J. Zhu (Editors)\hfill\break
Electronic Journal of Differential Equations, Conference~01, 1997, pp. 41--53.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break
ftp 147.26.103.110 or 129.120.3.113 (login: ftp)\bigskip} }
\topmatter
\title
PRACTICAL PERSISTENCE FOR DIFFERENTIAL
DELAY MODELS OF POPULATION INTERACTIONS
\endtitle
\thanks
{\it 1991 Mathematics Subject Classifications:} 34K25, 92D25.
\hfil\break\indent
{\it Key words and phrases:} Uniform persistence, practical persistence,
\hfil\break\indent
Kolmogorov population models, retarded functional differential equations.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and
University of North Texas. \hfil\break\indent
Published November 12, 1998.
\endthanks
\author Yulin Cao \& Thomas C. Gard \endauthor
\address Yulin Cao \hfill\break
Department of Mathematics \hfill\break
SUNY, College of Technology at Farmingdale \hfill\break
Farmingdale, NY 11735, USA.
\endaddress
\address Thomas C. Gard \hfill\break
Department of Mathematics \hfill\break
The University of Georgia \hfill\break
Athens, GA 30602, USA. \endaddress
\email gard\@math.uga.edu
\endemail
\abstract
Practical persistence refers to determining specific
estimates in terms of model data for the asymptotic distance to the
boundary of the feasible region for uniformly persistent population
interaction models. In this paper we illustrate practical
persistence by computing, using multiple Liapunov functions, such
estimates for a few basic examples of competition and predator-prey
type which may include time delays in the net per capita growth
rates.
\endabstract
\endtopmatter
\document
\head 1. Introduction \endhead
\noindent Uniform persistence for the Kolmogorov type models of
population interactions
$$
\dot{x}_i = x_i f_i(x)\quad (i = 1, \dots, n)\tag{1.1}
$$
means that solutions $x = x(t)$ of (1.1) which are initially
component-wise positive are asymptotically uniformly component-wise
positive: there are positive numbers $\delta_i$ such that if $x =
x(t) = \{x_i(t)\}$ is any solution of (1.1) with $x_i(0) > 0, i =
1, \dots, n$, then
$$
\liminf_{t \rightarrow +\infty} x_i(t) > \delta_i.\tag{1.2}
$$
\noindent In (1.1), $x_i(t)$ represents the population (density) of
the i-th species at time $t$ with its (total) net rate of growth given
by (1.1). Persistence for (1.1) corresponds to mutual survival for
the species represented in the model. Generally one expects
populations also to be bounded. More precisely, if also (1.1) is
(point) dissipative, i. e., if there are constants $M_i > 0$ such
that
$$
\limsup_{t\rightarrow +\infty} x_i(t) < M_i,\tag{1.3}
$$
then (1.1) is said to exhibit permanence. Uniform persistence (or
permanence) has emerged as an important stability concept for
population dynamics models (see Waltman [15], for example).
More generally, permanence indicates that a sustained level of
complexity is maintained in (1.1) in that at least the dimension of
the system is preserved for arbitrarily large time. The discussion
of persistence has been extended to differential equations in
infinite dimensional spaces including partial differential equations
and functional differential equations (Hale and Waltman [11], Hutson
and Schmitt [13]). The latter may involve time delays which
represent the extent of dependence on the past for solutions.
Delays can be discrete type $\tau_j$
$$
\dot{x}_i(t) = x_i(t) f_i(x(t), x(t - \tau_1), \dots, x(t -
\tau_m)) \quad (i = 1, \dots, n)\tag{1.4a}
$$
or continuous delays
$$
\dot{x}_i(t) = \sum^m_{j=1} \int^0_{-\tau_j} F_{ij}(x(t), x(t + s),
t, t + s)\ \! k_{ij}(s) \ \! ds\quad (i = 1, \dots, n)\tag{1.4b}
$$
where the $k_{ij}$ are distributions on the interval $(-\tau, 0],
\tau = \max \tau_j$. Also, as (1.4b) suggests, non-autonomous
systems can be addressed. For (1.4), initial conditions are
functions $\phi$ defined on the interval $(-\tau, 0]$:
$$
x(t) = \phi(t), t \in (-\tau, 0]\tag{1.5}
$$
and solutions are naturally considered as mappings on an appropriate
function space $Y((-\tau, 0])$
$$
x = x_t \in Y: x_t(\theta) = x(t + \theta), t \in (-\tau,
0].\tag{1.6}
$$
\noindent To obtain persistence, two techniques have been
employed: boundary-flow analysis and construction of Liapunov
functions. Here, the Liapunov functions are defined somewhat
differently from the classical definition in the equilibrium
stability setting. Sometimes referred to as an ``average''
Liapunov function (see [13] and the references therein) or a
``persistence'' function ([8]), this type of auxiliary function
indicates that the boundary of the (persisting) set repels the flow
defined by the differential equation inside the set. Generally
such a function is defined (and smooth) in a neighborhood of the
boundary, and, in the particular, it is continuous from inside the
set at the boundary. For multi-species population interactions
models (1.1), the basic choice for the function is
$$
V(x) = \prod^n_{i=1} x^{r_i}_i\tag{1.7}
$$
where $x = (x_1, \dots, x_n)$, and $r_1, \dots, r_n$ are positive
constants, and the set is the usual positive cone in $R^n$
$$
R^n_+ = \{ x = (x_1, \dots, x_n): x_i > 0, i = 1, \dots, n\}.
$$
In the approach which we take, the single function $V$ is replaced by a
number of functions which we call multiple Liapunov or net
functions and which satisfy less restrictive conditions than
above. The Liapunov function method, especially the variation involving
multiple Liapunov or net functions, allows determining practical
persistence ([4], [5]). Practical persistence (permanence) refers
to obtaining specific estimates for $\delta_i$ (and $M_i$) in terms
of the model data
$$
\delta_i = \delta_i(f)\quad (\text{and } M_i = M_i(f))\tag{1.8}
$$
such as, for example, in the case of simple food chains, in [7].
Such estimates can give indications whether persistence (and
dissipativity) are really meaningful for the model. There seems to
be some recent interest in extending the idea of practical
persistence to PDE ([1], [2]) and discrete population interaction
models ([12]). The main point of this paper is to illustrate
practical persistence as simply as possible by calculating the
$\delta$'s and $M$'s for some elementary 2-species models with
explicit self-limitation in each species and with or without a
single discrete time delay. We include a specific numerical
example - a Lotka-Volterra competition model with time delay
$\tau$. The model is globally stable for all $\tau \ge 0$, and so
an ideal estimate for practical persistence in this case should
involve specifying a small region in the positive quadrant
containing the stable equilibrium. The figure summarizing our
treatment of this example indicates how well we can achieve this
using a pair of simple Liapunov functions.
\bigskip
\noindent Generally, in the infinite dimensional case, the Liapunov
approach (see Freedman and Ruan[6], Hutson and Schmitt [13] and
Lakshmikantham and Matrosov [14]) has amounted to constructing
(average) Liapunov functionals on $Y$, or using the
Liapunov-Razumikhin technique ([10]) with Liapunov functions
defined on the range space
$X$ (the positive cone $R^n_+$ in $R^n$) for functions in $Y$.
In this setting, if $\partial X$ denotes the boundary of $X$,
uniform persistence means that there is a $\delta > 0$ such that if
$x(t)$ is any solution with $x(0) \in X\backslash
\partial X$, then
$$
\liminf_{t \rightarrow +\infty} d(x(t), \partial X) >
\delta\tag{1.9}
$$
where $d$ is the distance function in $X$. Here we construct a
set of multiple Liapunov functions
$$
\{V_1, \dots, V_p\}
$$
which are defined on possibly only a subset $X_0$ of $X$. If a
dissipative type property like (1.3) holds, a natural choice for
$X_0$ is generally suggested by the correspondingly bounded elements
of $X$. The main advantage of the multiple Liapunov function
approach is that the requirements are parceled out to several
functions on different portions of the set $X_0$, rather than a
single Liapunov function (or even a vector Liapunov function) on the
whole space $X$. Another way of looking at this scenario is that a
possibly complicated Liapunov function is being assembled
piecemeal. Our idea for this approach was originally motivated by
the paper of Wendi Wang and Ma Zhien [16]; indeed, our work ([3],
[4], [5]) on this problem amounts to a sequence of generalizations
and applications of the main result in [16]. This approach amounts
to constructing a partition $\{X_1, \dots, X_{p+1}\}$ of $X$ (or any
subset $X_0$ of $X$ which is the ultimate residence of all solution
trajectories) with the property that
$$
\text{dist }(X_{p+1}, \partial X) > 0.\tag{1.10}
$$
\noindent The sets $X_k$ in the partition are determined by the
functions $V_k$ and these sets are ordered by increasing time on
trajectories: for $k < p + 1$, trajectories in $X_k$ at some time
must leave $X_k$ in finite time and cannot move into $X_j$ for any
$j < k$ at any future time; consequently, ultimately all
trajectories lie in $X_{p+1}$. In the next section we give a concise
summary of our basic results on uniform persistence and practical
persistence for differential delay equation models of population
interactions. For simplicity and clarity we specialize our results
to 2-species interactions here. In section 3 we give a result
which determines specific dissipative bounds for 2-species models
with explicit self-limitation in each species and with or without a
single discrete time delay. We then discuss a numerical example - a
Lotka-Volterra competition model - to illustrate our practical
persistence result.
\heading 2. Persistence via multiple Liapunov functions
\endheading
\noindent We consider two-dimensional Kolmogorov type systems with
time delay
$$
\aligned
\dot{x}(t) &= x(t) f(x(t), y(t), x(t - \tau), y(t - \tau))\\
\dot{y}(t) &= y(t) g(x(t), y(t), x(t - \tau), y(t- \tau)).
\endaligned \tag{2.1}
$$
\noindent In (2.1), the functions $f$ and $g$ are continuous
functions defined on $\overline{R}^4_+$, the usual $4-d$ non-negative
cone. The result below is essentially a special case of the main
result appearing in [4]. Here we give a self-contained complete
proof of the result using two Liapunov functions.
\proclaim{Theorem 2.1} Suppose that:
\roster
\item"{(i)}" System (2.1) is dissipative with bounds $M_1$ and
$M_2$. \medskip
\item"{(ii)}" There are positive constants $\alpha$ and $\beta$
such that
$$
f(0, y_0, 0, y_1) - \alpha g(0, y_0, 0, y_1) > 0, \text{ all } y_0,
y_1 \in [0, M_2]\tag{2.2}
$$
and either
$$
\beta f(x_0, 0, x_1, 0) + g(x_0, 0, x_1, 0) > 0, \text{ all } x_0,
x_1 \in [0, M_1]\tag{2.3}
$$
or
$$
-\beta f(x_0, 0, x_1, 0) + g(x_0, 0, x_1, 0) > 0, \text{ all } x_0,
x_1 \in [0, M_1]\tag{2.4}
$$
\ \ \ \ provided $\alpha \beta < 1$.
\endroster
\noindent Then (2.1) is uniformly persistent (and hence permanent).
\endproclaim
\medskip
\demo{Proof of Theorem 2.1} With $M_1$ and $M_2$ as above, we denote
$$
X_0 = (0, M_1] \times (0, M_2].
$$
We will need the following constants to obtain a more
precise version of (ii):
$$
\aligned
P_1 &= \min\{f(x_0, y_0, x_1, y_1): ((x_0, y_0), (x_1, y_1)) \in
\overline{X}_0 \times \overline{X}_0\}\\
P_2 &= \min\{g(x_0, y_0, x_1, y_1): ((x_0, y_0), (x_1, y_1)) \in
\overline{X}_0 \times \overline{X}_0\}.
\endaligned \tag{2.5}
$$
$P_1$ and $P_2$ are finite by continuity of $f$ and $g$.
The dissipative property means that for any positive solution
$(x(t)$, $y(t))$ of (2.1), there is a $t_0 > 0$ such that
$$
(x(t), y(t)) \in X_0 \text{ for all } t > t_0.\tag{2.6}
$$
\noindent If $(x(t)$, $y(t))$ is a solution of (2.1) satisfying
$(x(t), y(t)) \in X_0$ for all $t \ge t^*_0 - 2\tau$ for some
$t^*_0 > 0$, then for any such $t$ and $0 < \nu < M_1$
$$
x(t) \le \nu \text{ implies } x(t -\tau) \le \nu e^{-P_1
\tau}.\tag{2.7}
$$
This follows because the integrated version of (2.1)
on the interval $[t - \tau, t]$ gives
$$
x(t) = x(t - \tau) \exp\int^t_{t-\tau} f(x(s), y(s), x(s - \tau),
y(s - \tau)) ds \ge x(t - \tau) e^{P_1\tau}
$$
for $t \ge t^*_0$. If $\displaystyle \nu^* = \nu e^{P_1 \tau}$, then
(2.7) becomes
$$
x(t) \le \nu^* \Rightarrow x(t - \tau) \le \nu\tag{2.8}
$$
and similarly for $y(t)$. Condition (ii) then implies, for
sufficiently small positive numbers $\epsilon_1$ and $\epsilon_2$,
there are positive numbers $\nu_1 < M_1$ and $\nu_2 < M_2$ such that
$$
\gather
f(x_0, y_0, x_1, y_1) - \alpha g(x_0, y_0, x_1, y_1) > \epsilon_1\\
\text{ for all } x_0 \in [0, \nu^*_1], x_1 \in [0, \nu_1]
\text{ and } y_0, y_1 \in [0, M_2]\\
\text{and}\tag{2.9}\\
\beta f(x_0, y_0, x_1, y_1) + g(x_0, y_0, x_1, y_1) > \epsilon_2\\
\text{for all } x_0, x_1 \in [0, M_1]
\text{ and } y_0 \in [0,
\nu^*_2], y_1 \in [0, \nu_2]
\endgather
$$
where
$$
\nu^*_1 = \nu_1 e^{P_1\tau} \text{ and } \nu^*_2 = \nu_2
e^{P_2\tau}.\tag{2.10}
$$
The proof of Theorem 2.1 makes use of the two Liapunov functions
$$
V_1 = xy^{-\alpha}\quad V_2 = x^{\beta} y.\tag{2.11}
$$
One asserts that there are positive constants $\eta_1$ and
$\eta_2$ such that the sets
$$
\align
X_1 &= \{(x, y) \in X_0: V_1(x, y) \le \eta_1\}\\
X_2 &= \{(x, y) \in X_0: V_2(x, y) \le \eta_2, V_1(x, y) >
\eta_1\}\tag{2.12}\\
X_3 &= \{(x, y) \in X_0: V_1(x, y) > \eta_1, V_2(x, y) > \eta_2\}
\endalign
$$
partition $X_0$ and structure the eventual location of trajectories
$(x(t), y(t))$ of positive solutions of (2.1) according to the
following scheme. Let $t_0$ be given by (2.6).
\roster
\item"{(P1)}" If $(x(t), y(t)) \in X_j$ for some $t_1 > t_0$ then
$$
(x(t), y(t)) \in \operatornamewithlimits\cup\limits^3_{i=j} X_i
\text{ for all } t > t_1.
$$
\item"{(P2)}" If $(x(t), y(t)) \in X_j$ for some $t_1 > t_0$
and $j < 3$, there is $t_2 > t_1$ such that
$$
(x(t_2), y(t_2)) \not\in X_j.
$$
\endroster
\noindent From (P1) and (P2) it follows that there is a $t_3$ such
that
$$
(x(t), y(t)) \in X_3 \text{ for all } t > t_3\tag{2.13}
$$
which gives uniform persistence. The $\delta$'s in the definition of
uniform persistence are determined from $\alpha$, $\beta$
and the $\eta$'s as follows:
$$
(x, y) \in X_3 \Rightarrow V_1 = xy^{-\alpha} > \eta_1 \text{ and }
V_2 = x^{\beta} y > \eta_2\tag{2.14}
$$
from which we obtain
$$
y > \eta_2/M^{\beta}_1 = \delta_2 \text{ and } x >
\eta_1(\eta_2/M^{\beta}_1)^{\alpha} = \delta_1\tag{2.15}
$$
i. e.,
$$
X_3 \subseteq (\delta_1, M_1] \times (\delta_2, M_2].\tag{2.16}
$$
\noindent It remains to obtain explicit expressions for $\eta_1$ and
$\eta_2$ in terms of $f$ and $g$ such that (P1) and (P2) hold. With
$\nu^*_1$ and $\nu^*_2$ as in (2.10), we choose
$$
\eta_1 = \nu^*_1/M^{\alpha}_2 \text{ and } \eta_2 = (\nu^*_2)^{1 +
\alpha \beta} \eta^{\beta}_1 = (\nu^*_2)^{1 +
\alpha\beta}(\nu^*_1/M^{\alpha}_2)^{\beta}.\tag{2.17}
$$
\noindent Now the first step toward establishing (P1) is to show
$$
X_1 \subseteq \{x \le \nu^*_1\} \text{ and } X_2 \subseteq \{y \le
\nu^*_2\}
\tag{2.18}
$$
with $\eta_1$ and $\eta_2$ given in (2.17). We verify the second
containment in (2.18):
$$
\gather
(x, y) \in X_2 \Leftrightarrow (x, y) \in X_0 = (0, M_1] \times (0,
M_2]\\
\text{ and } \tag{2.19}\\
V_2(x, y) = x^{\beta} y \le \eta_2,
V_1(x, y) = xy^{-\alpha} > \eta_1
\endgather
$$
and from (2.19) it follows that
$$
y \le \eta_2/x^{\beta} < \eta_2/(y^{\alpha}\eta_1)^{\beta}
\Leftrightarrow y^{1 + \alpha\beta} < \eta_2/\eta_1^{\beta}.
$$
Thus we have
$$
y < \left(\eta_2/\eta^{\beta}_1\right)^{1/(1 + \alpha\beta)} =
\nu^*_2.
$$
From (2.18) and (2.9) we conclude that, for any solution
$(x(t), y(t))$ of (2.1) which is in $X_1$ on the interval $[t - \tau,
t]$
$$
\aligned
\dot{V}_1 (x(t), y(t)) = \frac{d}{dt} V_1(x(t), y(t))&\\
=[f(x(t), y(t), x(t - \tau), &y(t - \tau)) - \alpha g(x(t), y(t),
x(t - \tau), y(t - \tau))] V_1(x(t), y(t))\\
\ge& \epsilon_1 V_1(x(t),y(t))
\endaligned \tag{2.20}
$$
and similarly
$$
\dot{V}_2 (x(t), y(t)) = \frac{d}{dt} V_2(x(t), y(t)) \ge
\epsilon_2V_2(x(t), y(t))\tag{2.21}
$$
if the solution $(x(t), y(t))$ is in $X_2$ on the interval $[t -
\tau, t]$. Now if (P1) fails, there is a $T > 0$ , an integer $k$
either 2 or 3, and a positive solution $(x(t)$, $y(t))$ of (2.1)
satisfying $(x(T), y(T)) \in X_k$ and for which
$$
t^* = \text{inf}\{t \ge T: (x(t), y(t)) \not\in
\operatornamewithlimits\cup\limits^3_{i=k} X_i\}\tag{2.22}
$$
is a well-defined finite number. Furthermore, by
continuity there is a positive integer $k^* < k$ such that
$$
(x(t^*), y(t^*)) \in X_k \text{ and } V_{k^*}((x(t^*), y(t^*)) =
\eta_{k^*}.\tag{2.23}
$$
\noindent Since $(x(T), y(T)) \not\in X_{k^*}, t^* > T$. From
either (2.20) or (2.21) (depending on whether $k^* =$ 1 or 2),
$V_{k^*}(x(t), y(t))$ is strictly increasing at $t = t^*$, and so
with (2.23) we have
$$
V_{k^*} (x(t), y(t)) < \eta_{k^*} , \text{ for } 0 < t^* - t <<
1.\tag{2.24}
$$
However, by definition of $t^*$ and $X_i$
$$
V_i(x(t), y(t)) > \eta_i , \text{ for all } t \in [T, t^*), i = 1,
\dots, k - 1\tag{2.25}
$$
which contradicts (2.24) since $k^*$ is one of these $i$'s. Thus
(P1) cannot fail. To verify (P2), we first note that since $X_1$,
$X_2$, and
$X_3$ partition $X_0$ and since (2.1) is dissipative, for any
positive solution $(x(t),y(t))$ of (2.1) there is a $T > 0$ , an
integer $k$ either 1, 2 or 3, such that
$$
(x(T), y(T)) \in X_k.\tag{2.26}
$$
\noindent It follows from (P1) that there is a $t_0 \ge T$ and an
integer $k_0, k \le k_0 \le 3$ such that
$$
(x(t), y(t)) \in X_{k_0} ,\text{ for all } t \ge t_0.\tag{2.27}
$$
If $k_0\ne 3$, then from (2.18)-(2.21), we have
$$
\frac{d}{dt} V_{k_0}(x(t), y(t)) \ge \epsilon_{k_0} V_{k_0}(x(t),
y(t)) , \text{ for all } t \in [t_0, \infty)
$$
which implies
$$
V_{k_0}(x(t), y(t)) \rightarrow \infty \text{ as } t \rightarrow
\infty
$$
and this contradicts $(x(t), y(t)) \in X_{k_0}$ for all
$t \ge t_0$. Thus (P2) is established. \qed
\enddemo
\noindent For practical persistence we will need specific choices
for $\nu_1$ and $\nu_2$. Since the magnitudes of $\epsilon_1$ and
$\epsilon_2$ are not important, by continuity we can choose $\nu_1$
and $\nu_2$ as large as possible with the property
$$
\gather
f(x_0, y_0, x_1, y_1) - \alpha g(x_0, y_0, x_1, y_1) > 0\\
\text{ for all } x_0 \in [0, \nu^*_1), x_1 \in [0, \nu_1)
\text{ and } y_0, y_1 \in [0, M_2]\\
\text{and}\tag{2.28}\\
\beta f(x_0, y_0, x_1, y_1 + g(x_0, y_0, x_1, y_1) > 0\\
\text{ for all } x_0, x_1 \in [0, M_1] \text{ and } y_0 \in [0,
\nu^*_2), y_1 \in [0, \nu_2)
\endgather
$$
with $\nu^*_1$ and $\nu^*_2$ as in (2.10):
$$
\nu^*_1 = \nu_1 e^{P_1\tau} \text{ and } \nu^*_2 = \nu_2
e^{P_2\tau}.\tag{2.29}
$$
\noindent Then (2.17) gives
$$
\eta_1 = \nu^*_1/M^{\alpha}_2 \text{ and } \eta_2 = (\nu^*_2)^{1 +
\alpha\beta}(\nu^*_1/M^{\alpha}_2)^{\beta}\tag{2.30}
$$
which in turn by (2.15) leads to
$$
\gather
\delta_1 = \eta_1(\eta_2/M^{\beta}_1)^{\alpha} =
\frac{\left(\nu^*_1 \nu^{*\alpha}_2\right)^{1 +
\alpha\beta}}{M^{\alpha\beta}_1(M^{\alpha}_2)^{1+ \alpha\beta}}\\
\text{and}\tag{2.31}\\
\delta_2 = \eta_2/M^{\beta}_1 = \frac{\nu^{*\beta}_1(\nu^*_2)^{1 +
\alpha\beta}}{M^{\beta}_1 M^{\alpha\beta}_2}.
\endgather
$$
\noindent If the second case of condition (ii) holds, i. e., (2.4)
replaces (2.3), we can use the two Liapunov
functions
$$
V_1 = xy^{-\alpha} \quad \ V_2 = x^{-\beta} y\tag{2.32}
$$
\noindent In this case
$$
(x, y) \in X_3 \Rightarrow V_1 = xy^{-\alpha} > \eta_1 \text{ and }
V_2 = x^{-\beta} y > \eta_2
$$
implies
$$
y > \left(\eta_2 \eta_1^{\beta}\right)^{1/(1 - \alpha\beta)} =
\delta_2
\text{ and } x > \eta_1 \delta^{\alpha}_2 = \delta_1,\tag{2.33}
$$
i.e.,
$$
X_3 \subseteq (\delta_1, M_1] \times (\delta_2, M_2].\tag{2.34}
$$
\noindent Here we require $\nu_1$, $\nu_2$, $\nu^*_1$ and $\nu^*_2$
such that
$$
\nu^*_1 = \nu_1 e^{P_1\tau} \text{ and } \nu^*_2 = \nu_2
e^{P_2\tau},\tag{2.35}
$$
$$
\gather
f(x_0, y_0, x_1, y_1) - \alpha g(x_0, y_0, x_1, y_1) > 0 \\
\text{ for all } x_0 \in [0, \nu^*_1), x_1 \in [0, \nu_1)
\text{ and } y_0, y_1 \in [0, M_2]\\
\text{and}\tag{2.36}\\
-\beta f(x_0, y_0, x_1, y_1) + g(x_0, y_0, x_1, y_1) > 0\\
\text{ for all } x_0, x_1 \in [0, M_1]
\text{ and } y_0 \in [0, \nu^*_2), y_1 \in [0, \nu_2).
\endgather
$$
\noindent Then with the choices
$$
\eta_1 = \nu^*_1/M^{\alpha}_2 \text{ and } \eta_2 =
\nu^*_2/M^{\beta}_1\tag{2.37}
$$
we obtain (2.18), and together with (2.33) we have
$$
\gather
\delta_2 = \left(\eta_2 \eta^{\beta}_1\right)^{1/(1 - \alpha\beta)}
=
\left(\frac{\nu^{*\beta}_1 \nu^*_2}{M^{\beta}_1
M^{\alpha\beta}_2}\right)^{1/(1 - \alpha\beta)}\\
\text{and}\tag{2.38}\\
\delta_1 = \eta_1 \delta^{\alpha}_2 =
\nu^*_1/M^{\alpha}_2\left(\frac{\nu^{*\beta}_1 \nu^*_2}{M^{\beta}_1
M^{\alpha\beta}_2}\right)^{\alpha/(1 - \alpha\beta)} =
\left(\frac{\nu^*_1 \nu^{*\alpha}_2}{M^{\alpha\beta}_1
M^{\alpha}_2}\right)^{1/(1 - \alpha\beta)}.
\endgather
$$
\noindent Estimates of practical persistence for (2.1) are provided
by (2.31) and (2.38). We summarize with a more specific version of
Theorem 2.1.
\proclaim{Theorem 2.2} (Practical Persistence) Assume the hypotheses
of Theorem 2.1. Let $\nu^*_1$ and $\nu^*_2$ be determined by (2.28)
and (2.29). If $(x(t),y(t))$ is any positive solution of (2.1),
there is a $t^* > 0$ such that
$$
(x(t), y(t)) \in [\delta_1, M_1] \times [\delta_2, M_2] \text{ for
all } t > t^*\tag{2.39}
$$
where $\delta_1$ and $\delta_2$ are given by (2.31). If condition
(2.3) in Theorem 2.1 is replaced by (2.4) and if $\nu^*_1$ and $\nu^*_2$
are given by (2.35)
and (2.36), then (2.39) holds with $\delta_1$ and $\delta_2$ are
given by (2.38).
\endproclaim
\heading 3. An example - Identical $L-V$ competitors
\endheading
\noindent We consider the Lotka-Volterra model for a pair
of identical competitors with time delay:
$$
\aligned
\dot{x}(t) &= x(t)\left[\frac{1}{2} - \frac{2}{3}x(t - \tau) -
\frac{1}{3}y(t - \tau)\right]\\
\dot{y}(t) &= y(t) \left[\frac{1}{2} - \frac{1}{3} x(t - \tau) -
\frac{2}{3}y(t - \tau)\right].
\endaligned \tag{3.1}
$$
\noindent For any time delay $\tau \ge 0$
$$
E = (x^*, y^*) = (.5, .5)
$$
is the unique equilibrium for (3.1) in $R^2_+$, and it is known (e.
g., see [9]) that $E$ is a globally (relative to $R^2_+$)
asymptotically stable for all $\tau \ge 0$. We would like our
practical persistence (permanence) estimates to be as close as
possible to the known attractor $E$ here. First we use the following
result from [4] to estimate dissipative constants $M_1 = M_2 = M$.
(For completeness, we include its proof in the appendix.)
\proclaim{Proposition 3.1} Suppose $u(t)$ is a $C^1$ positive
function defined on an interval $[t_0 - \tau, \infty)$ for some
$t_0 \ge 0$ which also satisfies the differential delay inequality
$$
\dot{u}(t) \le u(t) [a - bu(t - \tau)]\tag{3.2}
$$
where $a \ge 0$ and $b > 0$ are constants. Then
$$
\limsup_{t\rightarrow +\infty} u(t) \le \frac{a}{b}
e^{a\tau}.\tag{3.3}
$$
\endproclaim
\noindent Since $x$ and $y$ are non-negative (3.1) immediately gives
the uncoupled inequality system with identical components:
$$
\aligned
\dot{x}(t) \le x(t)\left[\frac{1}{2} - \frac{2}{3} x(t -
\tau)\right]\\
\dot{y}(t) \le y(t) \left[\frac{1}{2} - \frac{2}{3} y(t -
\tau)\right]
\endaligned \tag{3.4}
$$
and hence from Proposition 3.1
$$
M = M(\tau) = M_1(\tau) = M_2(\tau) = \frac{1/2}{2/3} e^{(1/2)\tau}
= .75 e^{.5\tau}.\tag{3.5}
$$
Positive solutions of (3.1) must eventually
reside in the square
$$
X_0 = (0, M] \times (0, M]
$$
and, corresponding to $\tau =$ 0, .25, and .5, for example, we have
$$
M = .75, .85 \text{ and } .97,\tag{3.6}
$$
respectively. Also, by symmetry here, we can take $\alpha = \beta$,
$\eta_1 = \eta_2$, $\nu_1 = \nu_2 = \nu$, and $P_1 = P_2 = P$. It
follows that $\nu^*_1 = \nu^*_2 = \nu^*$ and $\delta_1 = \delta_2 =
\delta$. We make use of the Liapunov functions
$$
V_1 = xy^{-.5}\quad V_2 = x^{-.5} y\tag{3.7}
$$
to investigate persistence. Our estimate for the attractor is the
set
$$
\aligned
X_3 &= \{(x, y) \in X_0: V_1(x, y) > \eta, V_2(x, y) > \eta\}\\
&= \{x \le M, y \le M, V_1(x, y) > \eta, V_2(x, y) > \eta\}
\endaligned \tag{3.8}
$$
where $M$ is given by (3.6) and $\eta$ is determined below. Toward
this end, according to (2.36), we first need $\nu$ (as large as
possible) with the property
$$
f(x, y) - \frac{1}{2} g(x, y) = \frac{1}{4} - \frac{x}{2} > 0
$$
for all $x \in [0, \nu)$ and $y \in [0, M]$. Thus $\nu =$ .5,
and we calculate, from (2.35),
$$
\nu^* = \nu e^{P\tau}\tag{3.9}
$$
where
$$
P = \min\{f(x, y): (x, y) \in \overline{X}_0\}.
$$
Here
$$
P = \frac{1}{2} - \frac{2}{3} M - \frac{1}{3}M = \frac{1}{2} - M =
-.25, -.35, \text{ and } -\!.47\tag{3.10}
$$
if $\tau =$ 0, .25, and .5, respectively, for example. Thus
$$
\nu^* = \nu e^{P\tau} = .5 e^{P\tau} = .5, .46, .40
$$
for $\tau =$ 0, .25, and .5 respectively, and finally we get from
(2.37)
$$
\eta = \nu^*/\sqrt{M} = .58, .50, .41.\tag{3.11}
$$
Finally, from (2.38)
$$
\delta = \left(\nu^{1.5}\right)^{1/(.75)} = \eta^2 = .34, .25,
.17.\tag{3.12}
$$
\input epsf.tex
\topinsert
\centerline{
\epsfxsize=11 true cm \epsfclipon
\epsffile{fig.eps}}
\centerline{ {\bf Figure 1.}}
\endinsert
Actually, for any $0 \le \tau < 1$ we have, from (3.6) and
(3.9)-(3.12),
$$
\delta = \frac{\nu^2 e^{(1 - 2M)\tau}}{M} = \frac{(.5)^2 e^{(1 - 1.5
e^{.5\tau})\tau}}{.75 e^{.5\tau}} \approx \frac{(1 - \tau)}{3}.
$$
\vskip 1.5cm
\heading{APPENDIX}\endheading
\demo{Proof of Proposition 3.1} We consider two cases - whether or
not $u(t)$ is eventually monotone. In the first case, if $u(t)$ is
eventually non-decreasing, say on the interval $(t_1, t_2)$, then
$u(t) \le a/b$ for $t > t_1 + \tau$, because $a - bu(t - \tau) \ge
0$ for $t > t_1$ from (3.2). If $u(t)$ is non-increasing on $(t_1,
\infty)$, then
$$
\lim_{t\rightarrow + \infty} u(t) = u_0
$$
is finite. If $u_0 > a/b$, then $u(t) > a/b + \epsilon$, some
$\epsilon > 0$ and $t >$ some $t_2 > t_1$. Thus for $t > t_2 +
\tau, u(t - \tau) > a/b + \epsilon$ and from (3.2)
$$
\dot{u}(t) \le u(t)[a - b(a/b + \epsilon)] = -b\ \! \epsilon\ \!
u(t).
$$
But this last inequality implies $u(t) \rightarrow 0$, as $t
\rightarrow \infty$, contradicting that $u_0 > a/b$. We conclude in
the monotone case that
$$
\limsup_{t\rightarrow +\infty} u(t) \le \frac{a}{b}.
$$
\noindent If $u(t)$ is not eventually monotone, $u(t)$ has a local
maximum at each point in a sequence $\{t_n\} \subseteq (t_0 + \tau,
\infty)$ with $t_n \rightarrow \infty$ as $n \rightarrow \infty$ and
$$
\limsup_{t\rightarrow +\infty} u(t)\quad = \lim_{n\rightarrow +
\infty} u(t_n).
$$
\noindent Since $\dot{u}(t_n) = 0$, (3.2) yields
$$
u(t_n - \tau) \le \frac{a}{b}.
$$
Certainly (3.2) gives $ \dot{u}(t) \le au(t)$
for all $t > t_0$. Thus integrating (3.2) on each interval
$[t_n - \tau, t_n]$ and using the previous two inequalities obtains
\medskip
$$
u(t_n) \le u(t_n - \tau)e^{a\tau} \le \frac{a}{b} e^{a\tau},
$$
and then
$$
\limsup_{t\rightarrow +\infty} u(t)\quad = \lim_{n\rightarrow +
\infty} u(t_n) \le \frac{a}{b} e^{a\tau}
$$
which completes the present proof.
\enddemo
\bigbreak
\heading REFERENCES\endheading \roster
\item"{[1]}" Cantrell, R. S., and C. Cosner, {\it Practical
persistence in ecological models via comparison methods}. Proc. Royal Soc.
Edinburgh 126A(1996), 247\--272.
\item"{[2]}" Cantrell, R. S., and C. Cosner, {\it Practical
persistence in diffusive food chain models}. Natural Resource Modeling
11(1998), 21\--34.
\item"{[3]}" Cao, Y., Fan, J.-P. and T. C. Gard, {\it Uniform
persistence for population models with time delay}. Applicable
Analysis {51}(1993), 197\--210.
\item"{[4]}" Cao, Y. and T. C. Gard, {\it Uniform persistence for
population models with time delay using multiple Liapunov
functions}. Differential and Integral Equations {6}(1993),
883\--898.
\item"{[5]}" Cao, Y. and T. C. Gard, {\it Uniform persistence and
net functions}. J. Dynamics and Differential Equations {7}(1995), 491\--520.
\item"{[6]}" Freedman, H. I. and S. Ruan, {\it Uniform persistence
in functional differential equations}. J. Differential Equations
{115}(1995), 173\--192.
\item"{[7]}" Gard, T. C., {\it Uniform persistence in multispecies
population models}. Math. Biosci. {85}(1987), 93\--104.
\item"{[8]}" Gard, T. C. and T. G. Hallam, {\it Persistence in food webs
- I. Lotka-Volterra food chains}. Bull. Math. Biol. 41(1979),
877\--891.
\item"{[9]}" Gopalsamy, K., {\it Time lags and global stability in
two-species competition}. Bull. Math. Biol. {42}(1980),
729\--737.
\item"{[10]}" Hale, J. K. and S. M. Verduyn Lunel, {\it Introduction
to Functional Differential Equations}. Springer-Verlag, New York,
1993.
\item"{[11]}" Hale, J. K. and P. Waltman, {\it Persistence in
infinite-dimensional systems}. SIAM J. Math. Anal. {20}(1989),
388\--395.
\item"{[12]}" Hutson, V. and K. Mischaikow, {\it An approach to
practical persistence}. preprint.
\item"{[13]}" Hutson, V. and K. Schmitt, {\it Permanence in
dynamical systems}. Math. Biosci. {111}(1992), 1\--71.
\item"{[14]}" Lakshmikantham, V. and V. M. Matrosov, {\it Vector
Liapunov Functions and Stability Analysis of Nonlinear Systems}.
Kluwer, Boston, 1991.
\item"{[15]}" Waltman, P., {\it A brief survey of persistence in
dynamical systems}. In: Delay Differential Equations and Dynamical
Systems, Lect. Notes in Math. 1475, 31\--40, Springer-Verlag, Berlin,
1991.
\item"{[16]}" Wang, W. and Z. Ma, {\it Harmless delays for uniform
persistence}. J. Math. Anal. Appl. {158}(1991), 256\--268.
\endroster
\enddocument