SUFFICIENT AND NECESSARY CONDITION FOR THE PERMANENCE OF PERIODIC PREDATOR-PREY SYSTEM

We consider the permanence of a periodic predator-prey system, where the prey disperse in a two-patch environment. We assume the Volterra within-patch dynamics and provide a sufficient and necessary condition to guarantee the predator and prey species to be permanent by using the techniques of inequality analysis. Our work improves previous relevant results.


Introduction.
Dispersal predator-prey systems described by autonomous ordinary differential equations have long played an important role in population biology (see [1,2,3,4,5,7,8,9,10,11,12,13,14,15,18,19,20,21,22,23,24] and the references cited therein).Recently, Lou and Ma [15] studied the following predator-prey system in two-patch environment: where x i (t) represents the prey population in the ith patch, i = 1, 2, at time t ≥ 0, y(t) stands for the predator population in patch 1 at time t ≥ 0; coefficients a i , b i , c i (i = 1, 2), d, l, and D are all positive constants.They proved that is a necessary and sufficient condition of the strong persistence of system (1.1),where (x * 1 (D), x * 2 (D)) is the globally asymptotically stable equilibrium of the following system: (1.3) Considering realistic models often requires the effects of the changing environment; we naturally expect that a similar condition should be selected for the permanence of the corresponding periodic predator-prey system, under the assumptions that the functions a i (t), b i (t), c i (t) (i = 1, 2), D(t), d(t), and l(t) are all positive, ω-periodic, and continuous for t ≥ 0.
Existing results on the permanence of system (1.4) have largely been restricted to some roughly sufficient conditions due to the increased complexity of global analysis for the nonautonomous systems (cf.Song and Chen [18]).The present paper provides a necessary and sufficient condition of the permanence of system (1.4) and removes some unnecessary conditions in [18].
The organization of this paper is as follows.In Section 2, we agree on some notations, give some definitions, and state three lemmas which will be essential to our proofs.In Section 3, by introducing the techniques found in [21], we obtain the necessary and sufficient condition which guarantees that system (1.4) is permanent.

Notations, definitions, and preliminaries.
In this section, we introduce some definitions and notations and state some results which will be useful in subsequent sections.Let C denote the space of all bounded continuous functions f : R → R, C 0 + the set of nonnegative f ∈ C, and C + the set of all f ∈ C such that f is bounded below by a positive constant.Given f ∈ C, we denote and define the lower average A L (f ) and upper average A M (f ) of f by is said to be permanent if there exists a compact set K in the interior of To prove the permanence of the species in (1.4), we need the information on the periodic logistic models with and without dispersal.Lemma 2.4 [25].The problem has exactly one canonical solution U if a ∈ C + , b ∈ C, and A L (b) > 0.Moreover, the following properties hold: For the dispersal logistic equations we have the following result.

Necessary and sufficient condition of permanence in (1.4) Theorem 3.1. System (1.4) is permanent if and only if
where (x * 1 (t), x * 2 (t)) is the globally asymptotically stable periodic solution of (2.10).
To prove this theorem, we need several propositions.In the rest of this paper, we denote by (x 1 (t), x 2 (t), y(t)) any solution of (1.4) with positive initial condition.Proposition 3.2.There exist positive constants M x and M y such that Proof.Obviously, R 3 + is a positively invariant set of (1.4).Given any positive solution (x 1 (t), x 2 (t), y(t)) of (1.4), we have ẋi on the other hand, the auxiliary equations have a unique globally asymptotically stable positive ω-periodic solution (x * 1 (t), x * 2 (t)).Let (u 1 (t), u 2 (t)) be the solution of (3.4) with u i (0) = x i (0).By Lemma 2.3, we have Moreover, from the global stability of (x * 1 (t), x * 2 (t)), for every given ε > 0, there exists T 0 > 0 such that In addition, for t ≥ T 0 , we have where y * (t) is the positive and globally asymptotically stable ω-periodic solution of the auxiliary logistic equation and M y = max 0≤t≤ω {y * (t) + ε}; then (3.2) holds for system (1.4).

Proposition 3.3. There exists a positive constant η x such that
(3.11) Proof.Suppose that (3.11) is not true; then there is a sequence where Choosing sufficiently small positive constants ε x and ε y such that ε x < 1, ε y < 1, and ) where 12), for the given ε x > 0, there exists a positive integer N 0 such that for m > N 0 .In the rest of this proof, we always assume that m > N 0 .The above inequality implies that there exists τ (m) 1 > 0 such that , and further . Let (u 1 (t), u 2 (t)) be any positive solution of the following auxiliary equations: for sufficiently large t > 0 and m > N 0 , which is a contradiction with (3.12).This completes the proof.} satisfying the following conditions: (3.28) (3.29)By Proposition 3.2, for a given integer m > 0, there is a according to the boundedness of ζ(t).By (3.13) and (3.14), there are constants P > 0 and N 0 > 0 such that for m ≥ N 0 , q ≥ K (m) , and a ≥ P .Inequality (3.36) implies for m ≥ N 0 , q ≥ K (m) .For the positive ε y satisfying (3.14) and (3.37), we have the following two circumstances: ,s (m) q + P ]; (ii) there exists τ ,s (m) q + P ] such that y(τ (m) q1 ,z m ) < ε y .
If (i) holds, by (3.38) we have which is a contradiction.If (ii) holds, we now claim that which is also a contradiction.This completes the proof.
Combining Propositions 3.2 to 3.6, we complete the proof of the sufficiency of Theorem 3.1.

Corollary 3.8. System (1.1) is permanent if and only if (1.2) holds.
This corollary implies that the strong persistence is equivalent to the permanence for system (1.1), and hence improves the main result (cf.[15,Theorem 2]).Remark 3.9.In [18], Song and Chen obtained that if the following conditions where d 0 is a positive number.By simple calculation, we have Hence (3.100) does not hold.We cannot get the permanence of (3.101) from the results of Song and Chen [18].However, we can obtain its permanence according to our result.In fact, from Lemma 2.5, we know that the following system, without a predator, has a positive periodic solution (x * 1 (t), x * 2 (t)) which is globally asymptotically stable.Denote (2π) −1 2π 0 x * 1 (t)dt = l 0 .Then l 0 is positive and condition (3.1) holds for d 0 < (1/2)l 0 .The permanence of (3.12) follows from Theorem 3.1, provided d 0 < (1/2)l 0 .Remark 3.11.Xu, Chaplain, and Davidson studied a more general model than (1.4) (see [22]) and provided the existence, uniqueness, and global stability of periodic solutions of the more general periodic predator-prey system.Conditions for uniform persistence are also stated.We note that their condition (H5) in [22] does not hold for a weak patchy environment (see [6]) in the sense that the intrinsic growth rate b i (t) may become negative on some time intervals.However, the discussion in this paper can be used to study the more reasonable weak patchy environment which is important for conservation of some endangered and rare species.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: m > ε y exp(αω).(3.41)By the continuity of y(t,z m ), there must exist τ q3 + (P (m) + 1)ω].By (3.13), we obtain

( 3 .
43) This contradiction establishes that (3.40) is true; particularly, (3.40) holds for t ∈ By (3.29) and (3.14), we have 2,...,n} such that all solutions starting in the interior of R n + ultimately enter and remain in K.The system is said to be strongly persistent if [17] for all solutions x(t) = (x 1 (t), x 2 (t),...,x n (t)) starting in the interior of R n + .xF(t,x) is the n × n matrix derivative of F with respect to x. Lemma 2.3[17].Let x(t) and y(t) be solution ofẋ = F(t,x), ẏ = G(t,y), (2.7) respectively, where both systems are assumed to have the uniqueness property for initial value problems.Assume both x(t) and y(t) belong to a domain D ⊂ R n for [t 0 ,t 1 ], in which one of the two systems is cooperative and ,T 5 +ω] as α → 0. Hence for ε 0 > 0, there exists positive α 0