\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. 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