Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni, & J. Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp. 41–53. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp 147.26.103.110 or 129.120.3.113 (login: ftp)

PRACTICAL PERSISTENCE FOR DIFFERENTIAL

DELAY MODELS OF POPULATION INTERACTIONS

Yulin Cao & Thomas C. Gard

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.

1. Introduction

Uniform persistence for the Kolmogorov type models of population interactions

ẋ_{i} = x_{i}f_{i}(x) (i = 1, …, n)  (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 δ_{i} such that if x = x(t) = {x_{i}(t)} is any solution of (1.1) with x_{i}(0) > 0, i = 1, …, n, then

lim inf_{t→+∞} x_{i}(t) > δ_{i}.  (1.2)

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

lim sup_{t→+∞} x_{i}(t) < M_{i},  (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 τ_{j}

1991 Mathematics Subject Classifications: 34K25, 92D25.

Key words and phrases: Uniform persistence, practical persistence,

Kolmogorov population models, retarded functional differential equations.

\begin{matrix}c \\ ◯ \end{matrix}1998 Southwest Texas State University and University of North Texas.

Published November 12, 1998.

ẋ_{i}(t) = x_{i}(t)f_{i}(x(t), x(t − τ_{1}), …, x(t − τ_{m})) (i = 1, …, n)  (1.4a)

or continuous delays

ẋ_{i}(t) = ∑_{j=1}^{m} ∫_{−τ_{j}}^{0} F_{ij}(x(t), x(t + s), t, t + s) k_{ij}(s) ds (i = 1, …, n)  (1.4b)

where the k_{ij} are distributions on the interval (−τ, 0], τ = max τ_{j}. Also, as (1.4b) suggests, non-autonomous systems can be addressed. For (1.4), initial conditions are functions ϕ defined on the interval (−τ, 0]:

x(t) = ϕ(t), t ∈ (−τ, 0]  (1.5)

and solutions are naturally considered as mappings on an appropriate function space Y ((−τ, 0])

x = x_{t} ∈ Y : x_{t}(θ) = x(t + θ), t ∈ (−τ, 0].  (1.6)

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) = ∏_{i=1}^{n} x_{i}^{r_{i}}  (1.7)

where x = (x_{1}, …, x_{n}), and r_{1}, …, r_{n} are positive constants, and the set is the usual positive cone in R^{n}

R_{+}^{n} = {x = (x_{1}, …, x_{n}) : x_{i} > 0, i = 1, …, 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 δ_{i} (and M_{i}) in terms of the model data

δ_{i} = δ_{i}(f) (and M_{i} = M_{i}(f))  (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 δ’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 τ. The model is globally stable for all τ ≥ 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.

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 ∂X denotes the boundary of X, uniform persistence means that there is a δ > 0 such that if x(t) is any solution with x(0) ∈ X\∂X, then

lim inf_{t→+∞} d(x(t), ∂X) > δ  (1.9)

where d is the distance function in X. Here we construct a set of multiple Liapunov functions

{V_{1}, …, 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}, …, 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

dist (X_{p+1}, ∂X) > 0.  (1.10)

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.

2. Persistence via multiple Liapunov functions

We consider two-dimensional Kolmogorov type systems with time delay

\begin{aligned}ẋ(t) & = x(t)f(x(t), y(t), x(t − τ), y(t − τ)) \\ ẏ(t) & = y(t)g(x(t), y(t), x(t − τ), y(t − τ)). \end{aligned}  (2.1)

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.

Theorem 2.1. Suppose that:

(i) System (2.1) is dissipative with bounds M_{1} and M_{2}.

(ii) There are positive constants α and β such that

f(0, y_{0}, 0, y_{1}) − αg(0, y_{0}, 0, y_{1}) > 0, all y_{0}, y_{1} ∈ [0, M_{2}]  (2.2)

and either

βf(x_{0}, 0, x_{1}, 0) + g(x_{0}, 0, x_{1}, 0) > 0, all x_{0}, x_{1} ∈ [0, M_{1}]  (2.3)

or

−βf(x_{0}, 0, x_{1}, 0) + g(x_{0}, 0, x_{1}, 0) > 0, all x_{0}, x_{1} ∈ [0, M_{1}]  (2.4)

provided αβ < 1.

Then (2.1) is uniformly persistent (and hence permanent).

Proof of Theorem 2.1. With M_{1} and M_{2} as above, we denote

X_{0} = (0, M_{1}] × (0, M_{2}].

We will need the following constants to obtain a more precise version of (ii):

\begin{aligned}P_{1} & = min\{f(x_{0}, y_{0}, x_{1}, y_{1}) : ((x_{0}, y_{0}), (x_{1}, y_{1})) ∈ \overline{X}_{0} × \overline{X}_{0}\} \\ P_{2} & = min\{g(x_{0}, y_{0}, x_{1}, y_{1}) : ((x_{0}, y_{0}), (x_{1}, y_{1})) ∈ \overline{X}_{0} × \overline{X}_{0}\}. \end{aligned}  (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)) ∈ X_{0} for all t > t_{0}.  (2.6)

If (x(t), y(t)) is a solution of (2.1) satisfying (x(t), y(t)) ∈ X_{0} for all t ≥ t_{0}^{∗} − 2τ for some t_{0}^{∗} > 0, then for any such t and 0 < ν < M_{1}

x(t) ≤ ν implies x(t − τ) ≤ νe^{−P_{1}τ} .  (2.7)

This follows because the integrated version of (2.1) on the interval [t − τ, t] gives

x(t) = x(t − τ) exp ∫_{t−τ}^{t} f(x(s), y(s), x(s − τ), y(s − τ))ds ≥ x(t − τ)e^{P_{1}τ}

for t ≥ t_{0}^{∗}. If ν^{∗} = νe^{P_{1}τ}, then (2.7) becomes

x(t) ≤ ν^{∗} ⇒ x(t − τ) ≤ ν  (2.8)

and similarly for y(t). Condition (ii) then implies, for sufficiently small positive numbers ϵ_{1} and ϵ_{2}, there are positive numbers ν_{1} < M_{1} and ν_{2} < M_{2} such that

f(x_{0}, y_{0}, x_{1}, y_{1}) − αg(x_{0}, y_{0}, x_{1}, y_{1}) > ϵ_{1}
for all x_{0} ∈ [0, ν_{1}^{∗}], x_{1} ∈ [0, ν_{1}] and y_{0}, y_{1} ∈ [0, M_{2}]
and  (2.9)
βf(x_{0}, y_{0}, x_{1}, y_{1}) + g(x_{0}, y_{0}, x_{1}, y_{1}) > ϵ_{2}
for all x_{0}, x_{1} ∈ [0, M_{1}] and y_{0} ∈ [0, ν_{2}^{∗}], y_{1} ∈ [0, ν_{2}]

where

ν_{1}^{∗} = ν_{1}e^{P_{1}τ} and ν_{2}^{∗} = ν_{2}e^{P_{2}τ} .  (2.10)

The proof of Theorem 2.1 makes use of the two Liapunov functions

V_{1} = xy^{−α} V_{2} = x^{β}y.  (2.11)

One asserts that there are positive constants η_{1} and η_{2} such that the sets

X_{1} = {(x, y) ∈ X_{0} : V_{1}(x, y) ≤ η_{1}}
X_{2} = {(x, y) ∈ X_{0} : V_{2}(x, y) ≤ η_{2}, V_{1}(x, y) > η_{1}}  (2.12)
X_{3} = {(x, y) ∈ X_{0} : V_{1}(x, y) > η_{1}, V_{2}(x, y) > η_{2}}

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).

(P1) If (x(t), y(t)) ∈ X_{j} for some t_{1} > t_{0} then

(x(t), y(t)) ∈ ∪_{i=j}^{3} X_{i} for all t > t_{1}.

(P2) If (x(t), y(t)) ∈ 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})) ∉ X_{j}.

From (P1) and (P2) it follows that there is a t_{3} such that

(x(t), y(t)) ∈ X_{3} for all t > t_{3}  (2.13)

which gives uniform persistence. The δ’s in the definition of uniform persistence are determined from α, β and the η’s as follows:

(x, y) ∈ X_{3} ⇒ V_{1} = xy^{−α} > η_{1} and V_{2} = x^{β}y > η_{2}  (2.14)

from which we obtain

y > η_{2}/M_{1}^{β} = δ_{2} and x > η_{1}(η_{2}/M_{1}^{β})^{α} = δ_{1}  (2.15)

i. e.,

X_{3} ⊆ (δ_{1}, M_{1}] × (δ_{2}, M_{2}].  (2.16)

It remains to obtain explicit expressions for η_{1} and η_{2} in terms of f and g such that (P1) and (P2) hold. With ν_{1}^{∗} and ν_{2}^{∗} as in (2.10), we choose

η_{1} = ν_{1}^{∗}/M_{2}^{α} and η_{2} = (ν_{2}^{∗})^{1+αβ}η_{1}^{β} = (ν_{2}^{∗})^{1+αβ}(ν_{1}^{∗}/M_{2}^{α})^{β}.  (2.17)

Now the first step toward establishing (P1) is to show

X_{1} ⊆ {x ≤ ν_{1}^{∗}} and X_{2} ⊆ {y ≤ ν_{2}^{∗}}  (2.18)

with η_{1} and η_{2} given in (2.17). We verify the second containment in (2.18):

(x, y) ∈ X_{2} ⇔ (x, y) ∈ X_{0} = (0, M_{1}] × (0, M_{2}]
and  (2.19)
V_{2}(x, y) = x^{β}y ≤ η_{2}, V_{1}(x, y) = xy^{−α} > η_{1}

and from (2.19) it follows that

y ≤ η_{2}/x^{β} < η_{2}/(y^{α}η_{1})^{β} ⇔ y^{1+αβ} < η_{2}/η_{1}^{β}.

Thus we have

y < (η_{2}/η_{1}^{β})^{1/(1+αβ)} = ν_{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 − τ, t]

V̇_{1}(x(t), y(t)) = \frac{d}{dt}V_{1}(x(t), y(t))

= [f(x(t), y(t), x(t − τ),y(t − τ)) − αg(x(t), y(t), x(t − τ), y(t − τ))]V_{1}(x(t), y(t))
≥ϵ_{1}V_{1}(x(t), y(t))

(2.20)

and similarly

V̇_{2}(x(t), y(t)) = \frac{d}{dt}V_{2}(x(t), y(t)) ≥ ϵ_{2}V_{2}(x(t), y(t))  (2.21)

if the solution (x(t), y(t)) is in X_{2} on the interval [t − τ, 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)) ∈ X_{k} and for which

t^{∗} = inf{t ≥ T : (x(t), y(t)) ∉ ∪_{i=k}^{3} X_{i}}  (2.22)

is a well-defined finite number. Furthermore, by continuity there is a positive integer k^{∗} < k such that

(x(t^{∗}), y(t^{∗})) ∈ X_{k} and V_{k^{∗}}((x(t^{∗}), y(t^{∗})) = η_{k^{∗}}.  (2.23)

Since (x(T), y(T)) ∉ 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)) < η_{k^{∗}}, for 0 < t^{∗} − t << 1.  (2.24)

However, by definition of t^{∗} and X_{i}

V_{i}(x(t), y(t)) > η_{i}, for all t ∈ [T, t^{∗}), i = 1, …, k − 1  (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)) ∈ X_{k}.  (2.26)

It follows from (P1) that there is a t_{0} ≥ T and an integer k_{0}, k ≤ k_{0} ≤ 3 such that

(x(t), y(t)) ∈ X_{k_{0}}, for all t ≥ t_{0}.  (2.27)

If k_{0} ≠ 3, then from (2.18)-(2.21), we have

\frac{d}{dt}V_{k_{0}}(x(t), y(t)) ≥ ϵ_{k_{0}}V_{k_{0}}(x(t), y(t)), for all t ∈ [t_{0}, ∞)

which implies

V_{k_{0}}(x(t), y(t)) → ∞ as t → ∞

and this contradicts (x(t), y(t)) ∈ X_{k_{0}} for all t ≥ t_{0}. Thus (P2) is established. □

For practical persistence we will need specific choices for ν_{1} and ν_{2}. Since the magnitudes of ϵ_{1} and ϵ_{2} are not important, by continuity we can choose ν_{1} and ν_{2} as large as possible with the property

f(x_{0}, y_{0}, x_{1}, y_{1}) − αg(x_{0}, y_{0}, x_{1}, y_{1}) > 0
for all x_{0} ∈ [0, ν_{1}^{∗}), x_{1} ∈ [0, ν_{1}) and y_{0}, y_{1} ∈ [0, M_{2}]
and  (2.28)
βf(x_{0}, y_{0}, x_{1}, y_{1} + g(x_{0}, y_{0}, x_{1}, y_{1}) > 0
for all x_{0}, x_{1} ∈ [0, M_{1}] and y_{0} ∈ [0, ν_{2}^{∗}), y_{1} ∈ [0, ν_{2})

with ν_{1}^{∗} and ν_{2}^{∗} as in (2.10):

ν_{1}^{∗} = ν_{1}e^{P_{1}τ} and ν_{2}^{∗} = ν_{2}e^{P_{2}τ} .  (2.29)

Then (2.17) gives

η_{1} = ν_{1}^{∗}/M_{2}^{α} and η_{2} = (ν_{2}^{∗})^{1+αβ}(ν_{1}^{∗}/M_{2}^{α})^{β}  (2.30)

which in turn by (2.15) leads to

δ_{1} = η_{1}(η_{2}/M_{1}^{β})^{α} = \frac{(ν_{1}^{∗}ν_{2}^{∗α})^{1+αβ}}{M_{1}^{αβ}(M_{2}^{α})^{1+αβ}}

and  (2.31)

δ_{2} = η_{2}/M_{1}^{β} = \frac{ν_{1}^{∗β}(ν_{2}^{∗})^{1+αβ}}{M_{1}^{β}M_{2}^{αβ}} .

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^{−α} V_{2} = x^{−β}y  (2.32)

In this case

(x, y) ∈ X_{3} ⇒ V_{1} = xy^{−α} > η_{1} and V_{2} = x^{−β}y > η_{2}

implies

y > (η_{2}η_{1}^{β})^{1/(1−αβ)} = δ_{2} and x > η_{1}δ_{2}^{α} = δ_{1},  (2.33)

i.e.,

X_{3} ⊆ (δ_{1}, M_{1}] × (δ_{2}, M_{2}].  (2.34)

Here we require ν_{1}, ν_{2}, ν_{1}^{∗} and ν_{2}^{∗} such that

ν_{1}^{∗} = ν_{1}e^{P_{1}τ} and ν_{2}^{∗} = ν_{2}e^{P_{2}τ} ,  (2.35)
f(x_{0}, y_{0}, x_{1}, y_{1}) − αg(x_{0}, y_{0}, x_{1}, y_{1}) > 0
for all x_{0} ∈ [0, ν_{1}^{∗}), x_{1} ∈ [0, ν_{1}) and y_{0}, y_{1} ∈ [0, M_{2}]
and  (2.36)
−βf(x_{0}, y_{0}, x_{1}, y_{1}) + g(x_{0}, y_{0}, x_{1}, y_{1}) > 0
for all x_{0}, x_{1} ∈ [0, M_{1}] and y_{0} ∈ [0, ν_{2}^{∗}), y_{1} ∈ [0, ν_{2}).

Then with the choices

η_{1} = ν_{1}^{∗}/M_{2}^{α} and η_{2} = ν_{2}^{∗}/M_{1}^{β}  (2.37)

we obtain (2.18), and together with (2.33) we have

δ_{2} = (η_{2}η_{1}^{β})^{1/(1−αβ)} = (\frac{ν_{1}^{∗β}ν_{2}^{∗}}{M_{1}^{β}M_{2}^{αβ}} )^{1/(1−αβ)}

and  (2.38)

δ_{1} = η_{1}δ_{2}^{α} = ν_{1}^{∗}/M_{2}^{α} (\frac{ν_{1}^{∗β}ν_{2}^{∗}}{M_{1}^{β}M_{2}^{αβ}} )^{α/(1−αβ)} = (\frac{ν_{1}^{∗}ν_{2}^{∗α}}{M_{1}^{αβ}M_{2}^{α}} )^{1/(1−αβ)} .

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.

Theorem 2.2. (Practical Persistence) Assume the hypotheses of Theorem 2.1. Let ν_{1}^{∗} and ν_{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)) ∈ [δ_{1}, M_{1}] × [δ_{2}, M_{2}] for all t > t^{∗}  (2.39)

where δ_{1} and δ_{2} are given by (2.31). If condition (2.3) in Theorem 2.1 is replaced by (2.4) and if ν_{1}^{∗} and ν_{2}^{∗} are given by (2.35) and (2.36), then (2.39) holds with δ_{1} and δ_{2} are given by (2.38).

3. An example - Identical L − V competitors

We consider the Lotka-Volterra model for a pair of identical competitors with time delay:

\begin{aligned}ẋ(t) & = x(t) [\frac{1}{2} − \frac{2}{3}x(t − τ) − \frac{1}{3}y(t − τ)] \\ ẏ(t) & = y(t) [\frac{1}{2} − \frac{1}{3}x(t − τ) − \frac{2}{3}y(t − τ)] . \end{aligned}  (3.1)

For any time delay τ ≥ 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 τ ≥ 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.)

Proposition 3.1. Suppose u(t) is a C^{1} positive function defined on an interval [t_{0} − τ, ∞) for some t_{0} ≥ 0 which also satisfies the differential delay inequality

u̇(t) ≤ u(t)[a − bu(t − τ)]  (3.2)

where a ≥ 0 and b > 0 are constants. Then

lim sup_{t→+∞} u(t) ≤ \frac{a}{b}e^{aτ} .  (3.3)

Since x and y are non-negative (3.1) immediately gives the uncoupled inequality system with identical components:

\begin{matrix}ẋ(t) ≤ x(t) [\frac{1}{2} − \frac{2}{3}x(t − τ)] \\ ẏ(t) ≤ y(t) [\frac{1}{2} − \frac{2}{3}y(t − τ)] \end{matrix}  (3.4)

and hence from Proposition 3.1

M = M(τ) = M_{1}(τ) = M_{2}(τ) = \frac{1/2}{2/3}e^{(1/2)τ} = .75e^{.5τ} .  (3.5)

Positive solutions of (3.1) must eventually reside in the square

X_{0} = (0, M] × (0, M]

and, corresponding to τ = 0, .25, and .5, for example, we have

M = .75, .85 and .97,  (3.6)

respectively. Also, by symmetry here, we can take α = β, η_{1} = η_{2}, ν_{1} = ν_{2} = ν, and P_{1} = P_{2} = P. It follows that ν_{1}^{∗} = ν_{2}^{∗} = ν^{∗} and δ_{1} = δ_{2} = δ. We make use of the Liapunov functions

V_{1} = xy^{−.5} V_{2} = x^{−.5}y  (3.7)

to investigate persistence. Our estimate for the attractor is the set

\begin{aligned}X_{3} & = \{(x, y) ∈ X_{0} : V_{1}(x, y) > η, V_{2}(x, y) > η\} \\ & = \{x ≤ M, y ≤ M, V_{1}(x, y) > η, V_{2}(x, y) > η\} \end{aligned}  (3.8)

where M is given by (3.6) and η is determined below. Toward this end, according to (2.36), we first need ν (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 ∈ [0, ν) and y ∈ [0, M]. Thus ν = .5, and we calculate, from (2.35),

ν^{∗} = νe^{P τ}  (3.9)

where

P = min\{f(x, y) : (x, y) ∈ \overline{X}_{0}\}.

Here

P = \frac{1}{2} − \frac{2}{3}M − \frac{1}{3}M = \frac{1}{2} − M = −.25, −.35, and −.47  (3.10)

if τ = 0, .25, and .5, respectively, for example. Thus

ν^{∗} = νe^{P τ} = .5e^{P τ} = .5, .46, .40

for τ = 0, .25, and .5 respectively, and finally we get from (2.37)

η = ν^{∗}/\sqrt{M} = .58, .50, .41.  (3.11)

Finally, from (2.38)

δ = (ν^{1.5})^{1/(.75)} = η^{2} = .34, .25, .17.  (3.12)

Figure 1.

Actually, for any 0 ≤ τ < 1 we have, from (3.6) and (3.9)-(3.12),

δ = \frac{ν^{2}e^{(1−2M)τ}}{M} = \frac{(.5)^{2}e^{(1−1.5e^{.5τ})τ}}{.75e^{.5τ}} ≈ \frac{(1 − τ)}{3}.

APPENDIX

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) ≤ a/b for t > t_{1} + τ, because a − bu(t − τ) ≥ 0 for t > t_{1} from (3.2). If u(t) is non-increasing on (t_{1}, ∞), then

lim_{t→+∞} u(t) = u_{0}

is finite. If u_{0} > a/b, then u(t) > a/b + ϵ, some ϵ > 0 and t > some t_{2} > t_{1}. Thus for t > t_{2} + τ, u(t − τ) > a/b + ϵ and from (3.2)

u̇(t) ≤ u(t)[a − b(a/b + ϵ)] = −b ϵ u(t).

But this last inequality implies u(t) → 0, as t → ∞, contradicting that u_{0} > a/b. We conclude in the monotone case that

lim sup_{t→+∞} u(t) ≤ \frac{a}{b}.

If u(t) is not eventually monotone, u(t) has a local maximum at each point in a sequence {t_{n}} ⊆ (t_{0} + τ, ∞) with t_{n} → ∞ as n → ∞ and

lim sup_{t→+∞} u(t) = lim_{n→+∞} u(t_{n}).

Since u̇(t_{n}) = 0, (3.2) yields

u(t_{n} − τ) ≤ \frac{a}{b}.

Certainly (3.2) gives u̇(t) ≤ au(t) for all t > t_{0}. Thus integrating (3.2) on each interval [t_{n} − τ, t_{n}] and using the previous two inequalities obtains

u(t_{n}) ≤ u(t_{n} − τ)e^{aτ} ≤ \frac{a}{b}e^{aτ} ,

and then

lim sup_{t→+∞} u(t) = lim_{n→+∞} u(t_{n}) ≤ \frac{a}{b}e^{aτ}

which completes the present proof.

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Yulin Cao Department of Mathematics SUNY, College of Technology at Farmingdale Farmingdale, NY 11735, USA.

Thomas C. Gard Department of Mathematics The University of Georgia Athens, GA 30602, USA.

E-mail address: gard@math.uga.edu