Predator-Prey Dynamics with Delay when Prey Dispersing in n-Patch Environment

Lin QIU and Taketomo MITSUI Graduate School of Human Informatics, Nagoya University, Nagoya 464-8601, Japan

February 8, 2002

Abstract

A model with time delay is considered for a predator-prey system. Here, a single species disperses between n patches of a heterogeneous environment with barriers between patches while a predator does not involve a barrier between patches. It is shown that the system is permanent under some appropriate conditions, and sufficient conditions are established for the global asymptotic stability of the positive equilibrium of the system.

Keywords: Permanence; globally asymptotically stable; predator-prey dynamics; equilibrium; positive solution.

1 Introduction

Much interest has been growing in the study of mathematical models of biological populations dispersing among patches in a heterogeneous environment, which has been the subject of several recent papers (see e.g. [1]–[7], [9, 10, 16, 17, 19, 20] and references cited therein). Particularly, the single population dispersing among patches has been studied in [1], [3]–[9], and the predator-prey interactions in a patchy environment have been dealt with in [2], [5]– [7], [10, 17, 18]. Some of them, [1, 4, 5, 17, 21, 22] deal with the question of global stability of the equilibrium solution.

This paper is concerned with a model of a single species that disperses among the n patches of a heterogeneous environment with barriers between patches, and with a predator against the species for which the dispersal between patches does not involve a barrier, and due to gestation the time delay is considered. Such models are often found in nature and we can find their examples in [11, 12].

The work in this paper can be regarded as a continuation of the work in [4, 5, 24]. In particular the model we choose to study here is based upon those developed in [5, 24]. Freedman and Takeuchi [5] considered the system of n + 1 autonomous ordinary differential equations as a model of the predator-prey system living in a patchy environment and the question of global stability for the single-species (prey) subsystem. They deal with the extinction and persistence of the predator. Wa and Ma [24] considered the asymptotic behavior of solutions of a predator-prey system incorporating time delay, in which the prey disperses between just two patches of a heterogeneous environment. Then how about the result for the model which deals with a single species that disperses between n patches?

In this study, we analyze such a system with n patches, each is permitted to have a different level of difficulty in its “escaper” barrier. Furthermore, once the population has left its present patch it may not successfully reach a new one of the environment (predation, harvesting, or for other reasons). In the analysis, we regard the probabilities of a successful transition between patches as a given condition, and show that the equilibrium is permanent under some appropriate conditions, and that the delay does not affect the permanence of the populations. Moreover, sufficient conditions are established for the global asymptotic stability of the positive equilibrium of the system.

The paper is organized as follows. In the next section, our model is described in detail. Section 3 deals with the question of the existence of positive equilibrium. In Section 4, we obtain sufficient conditions, under which the populations are permanent, and other sufficient conditions are established to ensure the absolute global asymptotic stability of the positive equilibrium of the system. A special case is considered in Section 5. Lastly, Section 6 gives a discussion of our results.

2 Modelling equations

We consider the following predator-prey system with diffusion and time delay in an n-patch environment:

\{ \begin{matrix}ẋ_{i} & = & x_{i}f_{i}(x_{i}, y) − ε_{i}h_{i}(x_{i}) + ∑_{j=1, j≠i}^{n}p_{ji}ε_{j}h_{j}(x_{j}), i = 1, 2, …, n \\ ẏ & = & y (−s(y) + ∑_{i=1}^{n}c_{i}P_{i}(x_{i}(t − τ))) , \end{matrix}  (2.1)

with x(θ) = ϕ(θ) ≥ 0, θ ∈ [−τ, 0], y(0) ≥ 0. Here the dot denotes the differentiation with respect to time, x_{i}(t) (i = 1, …, n) represents the prey population in the ith patch at a given time t ≥ −τ, while y(t) stands for the total predator population for n patches. The barriers of patches are assumed to be effective only as far as the prey population is concerned; thus the predator population has no barriers between patches. The function f_{i}(x, y) is the specific growth rate of the prey relating to the predator biomass y, the function h_{i}(x_{i}) is the desire to disperse out of the ith patch, p_{ji} is the probability of successful transition from jth patch to ith patch, where i is different from j, and P_{i}(x_{i}) is the predator functional response of the predator population on the prey in the ith patch. s(y) is the density-dependent death rate of the predator in the absence of its food (the prey).

The quantity τ ≥ 0 stands for a constant delay due to gestation. Moreover the inverse barrier strength ε_{j} is nonnegative, whereas the conversion ratio of prey into predator c_{i} positive. Fig. 2.1 displays the circumstances that the prey population, who leaves the jth patch, reachs other patches at time t. The hatched part in the jth patch is the total of the prey population leaving the patch at time t.

Figure 2.1: The n-patch configuration

Set

ℝ_{+}^{n} = {x = (x_{1}, …, x_{n}) : x_{i} ≥ 0, i = 1, …, n};

and let

C^{+} ≡ C([−τ, 0], ℝ_{+}^{n+1}),

denote the Banach space of continuous functions mapping the interval [−τ, 0] into ℝ_{+}^{n+1}. C^{+} is chosen as the initial function space for system (2.1).

The system (2.1) brings the following assumptions, all of which are usual in modelling the target phenomena. (H1) f_{i}, P_{i} and h_{i} are continuously differentiable for all i ; (H2) f_{i}(0, 0) > 0; ∂f_{i}/∂x_{i} < 0 in int ℝ_{+}^{n+1}; there is a k_{i} > 0 such that f_{i}(k_{i}, 0) = 0,

i = 1, …, n; (H3) ∂f_{i}/∂y < 0 in int ℝ_{+}^{n+1}, i = 1, …, n; (H4) P_{i}(0) = 0; P′_{i}(x) > 0 for positive x (i = 1, …, n); (H5) s(0) = 0, s′(y) > 0 for y > 0, s(y) → +∞ as y → +∞; (H6) h_{i}(0) = 0, η ≥ h′_{i}(x_{i}) ≥ h′_{i}(0) > 0;

(H7) 0 ≤ p_{ji} ≤ 1, ∑_{i=1, i≠j}^{n} p_{ji} ≤ 1.

Note that in (H6) the requirement h′_{i}(x) ≥ h′_{i}(0) > 0 is a technical condition required in the proofs. The quantity η is a bound on the growth rate of h_{i}(x_{i}) for all i. Without loss of generality, we assume that h′_{i}(0) = 1.

For notational simplicity, define p_{ii} = −1, i = 1, …, n and we rewrite the system (2.1) as

\{ \begin{matrix}ẋ_{i} & = & x_{i}f_{i}(x_{i}, y) + ∑_{j=1}^{n}p_{ji}ε_{j}h_{j}(x_{j}), i = 1, 2, …, n \\ ẏ & = & y (−s(y) + ∑_{i=1}^{n}c_{i}P_{i}(x_{i}(t − τ))) . \end{matrix}  (2.2)

Lastly, assume that

ε_{i} = α_{i}ε, α_{i} > 0, p_{ji} ≠ 0, i, j = 1, …, n,

then the system (2.2) becomes

\{ \begin{matrix}ẋ_{i} & = & x_{i}f_{i}(x_{i}, y) + ε∑_{j=1}^{n}p_{ji}α_{j}h_{j}(x_{j}), i = 1, 2, …, n \\ ẏ & = & y (−s(y) + ∑_{i=1}^{n}c_{i}P_{i}(x_{i}(t − τ))) , \end{matrix}  (2.3)

Hereafter we suppose that all the above assumptions are satisfied throughout the paper.

Since y(t) ≡ 0 satisfies the second equation of (2.3), when we denote by (ξ_{1}, ξ_{2}, …, ξ_{n}) the solution of the simultaneous equations

x_{i}f_{i}(x_{i}, 0) + ε∑_{j=1}^{n}p_{ji}α_{j}h_{j}(x_{j}) = 0 (i = 1, …, n)

with respect to x_{i} (i = 1, 2, …, n), the constant solution

x_{i}(t) ≡ ξ_{i} (i = 1, 2, …, n), y(t) ≡ 0

fulfils (2.3). Therefore, we introduce the quantity

d = −s(0) + ∑_{i=1}^{n}c_{i}P_{i}(ξ_{i}).  (2.4)

As d depends on ε, we often write d as d(ε), too.

Furthermore, let us introduce

ψ(x_{1}, …, x_{n}) = \{ \begin{matrix}s^{−1} (∑_{i=1}^{n}c_{i}P_{i}(x_{i})) & if ∑_{i=1}^{n}c_{i}P_{i}(x_{i}) > s(0), \\ 0 & otherwise, \end{matrix}

where s^{−1} denotes the inverse function of s(y), y ∈ [0, +∞), then ψ is readily proved to be nonnegative and continuous on ℝ_{+}^{n} and to satisfy

∂ψ/∂x_{i} > 0 i = 1, …, n, when ψ(x_{1}, …, x_{n}) > 0.

Next we will give the definition of notions which are fundamental in this paper.

Definition 2.1 ([16], P273) We say a population x(t) is permanent if there exist two positive constants m and M, m < M, such that, for sufficiently large t, the bound m ≤ x(t) ≤ M holds. We say a system of populations is permanent if all of its components are permanent.

Definition 2.2 ([16], P149) The equilibrium E(x_{1}, …, x_{n}, y) of system (2.1) (equivalently of (2.3)), if it exists, is said to be globally asymptotically stable (G.A.S.), if, for a fixed τ, all positive solutions of system (2.1) tend to E as t → +∞.

We say E(x_{1}, …, x_{n}, y) is absolutely globally asymptotically stable (A.G.A.S.) if it is globally asymptotically stable for all τ > 0.

For the discussion of the following sections, now we concern a general n-dimensional cooperative system

ẋ = F (x),  (2.5)

where F belongs to C^{1}-class on a domain ℝ_{+}^{n} and has Jacobian matrix DF (x) with nonnegative off-diagonal elements, i.e., for all i ≠ j, i, j = 1, …, n, ∂F_{i}/∂x_{j} ≥ 0, for all x ∈ ℝ_{+}^{n}. Denote the solution of (2.5) as x(t) whose initial value is x(0). The reference [13, 19, 20] gives the following.

Lemma 2.1 (i) Suppose that x(t) is the positive solution of (2.5), then x(t) is strictly increasing (decreasing) and converges to an equilibrium of (2.5) provided that F (x(0)) > 0 (F (x(0)) < 0) and it is bounded.

(ii) Suppose that x(t) is the positive solution of (2.5) existing on [0, +∞). Then if ṁ(t) ≤ F (m(t)), t ≥ 0, and m(0) ≤ x(0), we have m(t) ≤ x(t) for t ≥ 0. Furthermore, if ṁ(t) ≥ F (m(t)), t ≥ 0 and m(0) ≥ x(0), we have m(t) ≥ x(t), t ≥ 0.

3 Positive Equilibria

The origin E_{0}(0, …, 0) is clearly an equilibrium of (2.3). When ε = 0, there may be an equilibrium in the positive subspace, i.e., of the form Ê(ξ_{1}, …, ξ_{n}, 0), where ξ_{i} > 0,i = 1, …, n. Let P denote the n × n matrix (p_{ji}). By [4], we know that Ê also exists for sufficiently small ε, and they show that if det P = 0, Ê exists for all ε > 0. Hereafter, we deal with the positive equilibrium of the system (2.1).

We fix y as a nonnegative λ and consider the subcommunity equations consisting of n preys:

ẋ_{i} = x_{i}f_{i}(x_{i}, λ) + ε∑_{j=1}^{n}p_{ji}α_{j}h_{j}(x_{j}) (i = 1, …, n).  (3.1)

Let (x_{1}^{∗}(λ), …, x_{n}^{∗}(λ)) be the positive equilibrium of (3.1).

We will study the global stability of the positive equilibrium, when it exists, of system (3.1) and denote this equilibrium by E_{(λ)}(x_{1}^{∗}(λ), …, x_{n}^{∗}(λ)) throughout this section. We assume that E_{(λ)} is unique if it exists.

Lemma 3.1 Suppose that λ satisfies f_{i}(0, λ) > 0 (i = 1, …, n), and the two conditions

(i) 0 ≤ ε ≤ min(f_{i}(0, λ)/α_{i}) (i = 1, …, n),

(ii) lim_{x_{i}→∞} f_{i}(x_{i}, y) = −∞

hold. Then the system (3.1) has a G.A.S. positive equilibrium E_{(λ)}.

One can adopt the technique of Theorems 3.1 and 4.1 in [5] to prove this lemma, since the system (3.1) is similar to that given by Eq. (3.4) of [5].

Lemma 3.2 Suppose the assumptions of Lemma 3.1 are satisfied, then every component of the equilibrium E_{(λ)} of system (3.1) is continuous and strictly decreasing with respect to λ defined on the set D = {λ ≥ 0 : f_{i}(0, λ) > 0 (i = 1, …, n)}.

Proof. Let λ_{1} and λ_{2} be in D and ordered as λ_{1} < λ_{2}. Suppose that (u_{1}(t), …, u_{n}(t)) is the solution of the system (3.1) initiating at (x_{1}^{∗}(λ_{2}), …, x_{n}^{∗}(λ_{2})).

Now we rewrite the subcommunity equations (3.1) as

ẋ_{i} = F_{i}(x_{1}, …, x_{n}; λ), i = 1, …, n.  (3.2)

The assumption (H3) implies the inequality

F_{i}(u_{1}(0), …, u_{n}(0); λ_{1}) > F_{i}(x_{1}^{∗}(λ_{2}), …, x_{n}^{∗}(λ_{2}); λ_{2}) = 0 (i = 1, …, n),

which asserts that (u_{1}(t), …, u_{n}(t)) is strictly increasing and converges to (x_{1}^{∗}(λ_{1}), …, x_{n}^{∗}(λ_{1})) by applying Lemma 2.1. Consequently the order x_{i}^{∗}(λ_{1}) > x_{i}^{∗}(λ_{2}) holds for all i = 1, …, n. Therefore (x_{1}^{∗}(λ), …, x_{n}^{∗}(λ)) is strictly decreasing.

Next we will show the continuity. Let

x^{∗}(λ) = E_{(λ)}(x_{1}^{∗}(λ), …, x_{n}^{∗}(λ))

be a positive equilibrium of the system (3.1). For a function defined by

φ_{i}(a, x^{∗}) = ax_{i}^{∗}f_{i}(ax_{i}^{∗}, λ) − εα_{i}h_{i}(ax_{i}^{∗}) + ε ∑_{j=1, j≠i}^{n}p_{ji}α_{j}h_{j}(ax̂_{j}),

we have

\frac{∂φ_{i}}{∂a} (0, x^{∗}) = x_{i}^{∗}f_{i}(0, λ) − x_{i}^{∗}εα_{i} + ε ∑_{j=1, j≠i}p_{ji}α_{j}x_{j}^{∗}.

Since 0 ≤ ε ≤ min_{1≤i≤n}(f_{i}(0, λ)/α_{i}), we obtain

\frac{∂φ_{i}}{∂a} (0, x^{∗}) > 0 for i = 1, …, n.

Hence, there exists a positive a_{0} which gives φ(a, x̂(λ)) > 0 for 0 < a < a_{0}, i = 1, …, n. Since ∂f_{i}/∂x_{i} < 0 and h′_{i} is bounded, the equation lim_{x_{i}→∞} f_{i}(x_{i}, y) = −∞ yields that there

exists a positive b_{0} which derives

φ_{i}(b, x^{∗}) < 0 for b > b_{0} (i = 1, …, n).

Henceforth we obtain

F_{i}(a_{0}x_{1}^{∗}(λ), …, a_{0}x_{n}^{∗}(λ), λ) > 0, F_{i}(b_{0}x_{1}^{∗}(λ), …, b_{0}x_{n}^{∗}(λ), λ) < 0

and

a_{0}x_{i}^{∗}(λ) < x_{i}^{∗}(λ_{2}) < x_{i}^{∗}(λ_{1}) < b_{0}x_{i}^{∗}(λ)

for i = 1, …, n and λ_{1} ≤ λ ≤ λ_{2}. Let â_{i} = a_{0}x_{i}^{∗}(λ) and b̂_{i} = b_{0}x_{i}^{∗}(λ) (i = 1, …, n), and introduce a point-set E of ℝ_{+}^{n} as

E = {(x_{1}, …, x_{n}) : â_{i} < x_{i} < b̂_{i}, i = 1, …, n, }.

Then it is easily verified that E is a positive invariant of the system (3.1).

Let x(t; x_{0}, λ) be the positive solution of (3.1) initiating at x_{0}. Since (3.1) is autonomous, it is easily shown that (x_{1}^{∗}(λ^{∗}), …, x_{n}^{∗}(λ^{∗})) is uniform attractor of the positive solutions of (3.1), where λ = λ^{∗}. Consequently, for any β > 0, there is a T > 0 such that any solution x(t; x_{0}, λ^{∗}) of (3.1) starting with x_{0} ∈ E lies in the β-neighbourhood of (x_{1}^{∗}(λ^{∗}), …, x_{n}^{∗}(λ^{∗})) for t > T.

Furthermore, one can choose a positive δ such that x(T, x_{0}, λ) lies in the (2β)-neighbourhood of (x_{1}^{∗}(λ^{∗}), …, x_{n}^{∗}(λ^{∗})) when |λ − λ^{∗}| < δ, for the solutions depend on the parameter λ continouously. As a result, (x_{1}^{∗}(λ), …, x_{n}^{∗}(λ)) lies in (2β)-neighbourhood for |λ − λ^{∗}| < δ. This proves that the x_{i}^{∗}(λ) (i = 1, …, n) are continuous with respect to λ.

Lemma 3.3 Suppose the assumption (ii) of Lemma 3.1 are satisfied and the following conditions hold.

(1) −s(0) + ∑_{i=1}^{n}c_{i}P_{i}(x_{i}^{∗}(0)) > 0;

(2) there exists a positive λ^{∗} satisfying

min_{1≤i≤n} f_{i}(0, λ^{∗}) > 0 and lim_{λ→λ^{∗}−0}{−s(λ) + ∑_{i=1}^{n}c_{i}P_{i}(x_{i}(λ^{∗}))} < 0.

Then there exists a unique positive equilibrium (e_{1}, …, e_{n+1}) of system (2.3) with 0 ≤ ε ≤ min_{1≤i≤n}(f_{i}(0, λ^{∗})/α_{i}) in the region 0 < y ≤ λ^{∗}.

Proof. As is described at the beginning of this section, the positive equilibrium of the system (2.1) can be determined by the following system of equations:

H_{i}(x_{1}, …, x_{n}, y) = 0 (i = 1, …, n), and y = ψ(x_{1}, …, x_{n}),  (3.3)

where H_{i}(x_{1}, …, x_{n}, y) = x_{i}f_{i}(x_{i}, λ) + ε∑_{j=1}^{n}p_{ji}α_{j}h_{j}(x_{j}). Since f_{i}(0, y) > 0 for 0 < y ≤ λ^{∗}

(i = 1, …, n), Eq. (3.3) has a unique solution (x_{1}^{∗}(y), …, x_{n}^{∗}(y)) when 0 < y ≤ λ^{∗} and 0 ≤ ε ≤ min_{1≤i≤n}(f_{i}(0, λ^{∗})/α_{i}). Due to Lemma 3.2 x_{i}^{∗}(y) is strictly decreasing with respect to

y for every i. Therefore the function G(y) = y − ψ(x_{1}^{∗}(y), …, x_{n}^{∗}(y)) is strictly increasing. The inequalities G(0) < 0 and lim_{λ→λ^{∗}−0} G(λ) > 0 imply the unique existence of the positive solution for G(y) = 0 not exceeding λ^{∗}.

4 Criteria for Stability

The main purpose of this section is to show under which conditions the system is permanent and the positive equilibrium of system (2.1) is globally asymptotically stable.

Theorem 4.1 Let the equilibrium Ẽ of the system (2.3) exists uniquely, and E_{(0)}(x_{1}^{∗}(0), …, x_{n}^{∗}(0)) is G.A.S. in the x-space. Then the negativeness of d = d(ε) defined by (2.4) implies the limits lim_{t→+∞} x_{i}(t) = x_{i}^{∗}(0) (i = 1, …, n) and lim_{t→+∞} y(t) = 0.

Proof. Consider the system

ẋ_{i} ≤ x_{i}f_{i}(x_{i}, 0) + ε∑_{j=1}^{n}p_{ji}α_{j}h_{j}(x_{j}), i = 1, …, n,  (4.1)

and compare this with

v̇_{i} = v_{i}f_{i}(v_{i}, 0) + ε∑_{j=1}^{n}p_{ji}α_{j}h_{j}(v_{j}), i = 1, …, n.  (4.2)

Since the solutions of system (4.2) tend to E_{(0)} as t → ∞ by the hypothesis, there exists a positive T giving the inequality

x_{i}(t) ≤ x_{i}^{∗}(0) + δ for t ≥ T.

Choose δ > 0 so small that the inequality −s(0) + ∑_{i=1}^{n}c_{i}P_{i}(x_{i}^{∗}(0) + δ) < 0 holds. Then we

have ẏ < y[−s(0) + ∑_{i=1}^{n}c_{i}P_{i}(x_{i}(t − τ))] < 0 for t ≥ T + τ. This completes the proof.

Theorem 4.2 Suppose d(ε) > 0. Then the system (2.1) exhibits permanence.

Proof. At first, we show the positive solutions of system (2.1) are eventually bounded. Since the solutions are positive and we assume (H3), we have

\begin{aligned}ẋ_{i} & = & x_{i}f_{i}(x_{i}, y) + ε∑_{j=1}^{n}p_{ji}α_{j}h_{j}(x_{j}) \\ & < & x_{i}f_{i}(x_{i}, 0) + ε∑_{j=1}^{n}p_{ji}α_{j}h_{j}(x_{j}). i = 1, …, n. \end{aligned}

Comparing this system with

ẋ_{i} = x_{i}f_{i}(x_{i}, 0) + ε∑_{j=1}^{n}p_{ji}α_{j}h_{j}(x_{j}), i = 1, …, n,

we can find (b_{1}, …, b_{n}) with positive b_{i} (i = 1, …, n) which gives upper bound of any positive solution (x_{1}(t), …, x_{n}(t), y(t)) of system (2.1) as x_{i}(t) < b_{i} (i = 1, …, n) for sufficiently large t. This can be seen though the proof of Lemma 3.2 by letting, for example, b_{i} = b_{0}x_{i}^{∗}(0).

Henceforth the first n components of positive solutions of system (2.1) are eventually bounded, and thus, the sum of c_{i}P_{i}(x_{i}(t − τ)), (i = 1, …, n), is eventually bounded as well. By (H5), there exists k > 0 such that y(t) < k for all sufficiently large t. Therefore the positive solutions of system (2.1) are eventually bounded.

Secondly, we show uniformly persistence of the system. Let

\begin{aligned}C_{1} & = & \{ζ = (0, …, 0, ζ_{n+1}) ∈ C^{+}\}; \\ C_{2} & = & \{ζ = (ζ_{1}, …, ζ_{n}, 0) ∈ C^{+}\}. \end{aligned}

And let C^{∗} = C_{1} ∪ C_{2}, by Theorem 4.1 in [8], and adopting the technique of Theorem 1 in [24], we can know C^{∗} is a uniform repeller. Therefore, there exists a δ_{1} such that for any positive solution (u_{1}(t), …, u_{n}(t), v(t)) of system (2.1), v(t) > δ_{1} for all sufficiently large t and {u_{1}(t), …, u_{n}(t), v(t)} lies outside of the region

{(x_{1}, …, x_{n}, y) ∈ ℝ_{n+1}^{+} : 0 ≤ x_{i} ≤ δ_{1}, i = 1, …, n},

when t is sufficiently large.

Without loss of generality, we can assume that there exists an integer m_{1} (1 ≤ m_{1} < n) satisfying

0 ≤ x_{i} ≤ δ_{i}, i = 1, …, m_{1}
δ_{j} ≤ x_{j} ≤ b_{j}, j = m_{1} + 1, …, n,
0 < y < k,

with positive δ_{i} (i = 1, …, n).

Then we have

\begin{aligned}ẋ_{i} & = & x_{i}f_{i}(x_{i}, y) − εα_{i}h_{i}(x_{i}) + ε∑_{j≠i}p_{ji}α_{j}h_{j}(x_{j}) \\ & ≥ & x_{i}f_{i}(x_{i}, y) − εα_{i}h_{i}(x_{i}) + ε ∑_{j=m_{1}+1}^{n}p_{ji}α_{j}h_{j}(δ_{j}), \end{aligned}

for i = 1, …, m_{1}.

Since there exist δ_{i}^{∗} and δ̄_{i} satisfying both conditions

h_{i}(x_{i}) < \frac{1}{2α_{i}} ∑_{j=m_{1}+1}^{n}p_{ji}α_{j}h_{j}(δ_{j}) for 0 ≤ x_{i} ≤ δ_{i}^{∗}, i = 1, …, m_{1},

and

f_{i}(x_{i}, y) > 0 for 0 < y ≤ δ̄_{i}, 0 ≤ x_{i} ≤ δ̄_{i}, i = 1, …, m_{1},

by putting

δ̃_{i} = min(δ̄_{i}, δ_{i}^{∗}),

we obtain

ẋ_{i} > \frac{ε}{2} ∑_{j=m_{1}+1}^{n}p_{ji}α_{j}h_{j}(ε_{j}), if 0 ≤ x_{i} ≤ δ̃_{i}.

Therefore the positive solution (u_{1}(t), …, u_{n}(t), v(t)) can not enter into the region

{x ∈ ℝ_{n+1}^{+}, 0 ≤ x_{i} ≤ δ_{i}, (i = 1, …, m_{1}),
δ_{j} ≤ x_{j} ≤ b_{j}, (j = m_{1} + 1, …, n),
0 < y ≤ k},

when t is sufficiently large, for any m_{1} (1 ≤ m_{1} < n).

The above discussion leads to the

u_{i}(t) > δ_{i}, for all large t and i = 1, …, n,

which completes the proof.

Theorem 4.3 Provided that we have the following conditions

(a) d(ε) > 0,

(b) f_{i}(0, h) > 0, for i = 1, …, n, 0 ≤ ε ≤ min_{1≤i≤n}(f_{i}(0, h)/α_{i}), where h = ψ(x_{1}^{∗}(0), …, x_{n}^{∗}(0)),

(c) lim_{x_{i}→+∞} f_{i}(x_{i}, y) = −∞,

then the system (2.1) has a unique positive equilibrium (e_{1}, …, e_{n+1}).

The proof can be obtained by the technique of Theorem 2 in [24].

And Theorem 3 and Corollary 4 in [24] readily yield the following two results of this section.

Theorem 4.4 Under the assumptions of Theorem 4.3, the positive equilibrium (e_{1}, …, e_{n+1}) is absolutely globally asymptotically stable, provided that

σ(σ(y)) < y for e_{n+1} < y < h,  (4.3)

where σ(y) = ψ(x_{1}^{∗}(y), …, x_{n}^{∗}(y)).

Corollary 4.1 Suppose the assumptions of Theorem 4.4 are satisfied except that the condition on σ is replaced with

−1 < σ′(x) ≤ 0 for all x ∈ (0, h].

Then the statement of Theorem 4.4 remains valid.

5 The special case of linear h_{i}(x_{i})

In this section, we deal with the special case that the h_{i}(x_{i}) are linear. Then the system (2.1) can be converted to

\begin{aligned}ẋ_{i} & = & x_{i}f_{i}(x_{i}, y) + ε∑_{j=1}^{n}p_{ji}β_{j}x_{j}, \\ ẏ & = & y (−s(y) + ∑_{i=1}^{n}c_{i}P_{i}(x_{i}(t − τ))) . \end{aligned}  (5.1)

We show that the theorems of the previous section are valid without the requirement of the assumption for ε and the assumption (ii) in Lemma 3.1.

When y(t) = λ, the function φ_{i} in the proof of Lemma 3.2 can be simplified as

φ_{i}(a, x̂) = ax̂_{i}f_{i}(ax̂_{i}, λ) + aε∑_{j=1}^{n}p_{ji}β_{j}x̂_{j} = ax̂_{i}(f_{i}(ax̂_{i}, λ) − f_{i}(x̂_{i}, λ)).

Since ∂f_{i}/∂x_{i} is negative, and f_{i}(x_{i}, λ) is a decreasing function of x_{i}, we can get

φ_{i}(a, x̂) < 0 for 0 < a < 1.

On the other hand, we have

φ_{i}(a, x̂) > 0 for a > 1.

Therefore we can assert

F_{i}(ax̂_{1}, …, ax̂_{n}, λ) < 0 for 0 < a < 1 and F_{i}(ax̂_{1}, …, ax̂_{n}, λ) > 0 for a > 1,

where F_{i} is defined by (3.2).

Due to Theorem 4.4 in [4], the equality det P = 0 implies that the system (3.1) has an equilibrium, which is shown to be A.G.A.S. by Theorem 6.1 in [5]. Therefore, applying similar techniques in the proofs of the theorems of the present paper, we can obtain the following.

Theorem 5.1 If det P = 0, d(ε) > 0 and the condition (4.3) hold, then the system (2.1) with linear h_{i}(x_{i}) (i = 1, …, n) has a unique A.G.A.S. equilibrium for all ε ≥ 0.

Remark 5.1 In the case of linear h_{i}(x_{i}), the assumption of the uniqueness of E_{(λ)} in Section 3 can be relaxed, for Theorem 6.2 of [5] guarantees the uniquness as an automatic consequence of the linearity.

Remark 5.2 Consider the case that when any member of the prey leaves a given patch, it successfully reach a new patch in the environment. As described in [11], the case is observed in the mite dispersal on strawberry patches. Then the condition det P = 0 is always satisfied, since ∑_{i=1,i≠j}^{n} p_{ji} = 1. Hence the assumptions of Theorem 5.1 can be simplified.

6 Concluding remarks

We have analyzed a predator-prey system with time delay and dispersal among n patches in a heterogeneous environment. The barriers are considered only for the prey, but not for the predator, between patches. Furthermore, we have assumed that there is a positive probability that any member of the prey, who leaves a given patch, may not reach safely any other patch in the environment. Criteria for permanence and extinction are presented. Conditions, under which the positive steady state is absolutely globally asymptotically stable, are obtained by the continuity and monotonicity of the subsystem of the preys. Furthermore we have obtained criteria for the equilibrium to be absolutely globally asymptotically stable in the case that the dispersal is linear.

Theorem 4.2 may be regarded as an extension of similar results in [5] and [24]. In the former they show the persistence of the system without delay term, which means that each component N(t) of the system satisfies lim inf_{t→∞} N(t) > 0, when N(0) > 0. The latter considered a model in a two-patch environment and with time delays. Since we deal with the delay in the system, the system (2.1) is obviously more general than either in [24] or in [5].

Mathematically, we can further assume that there are barriers against predator dispersal among patches. We leave this to future studies.

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