Asymptotical Behaviors of Nonautonomous Discrete Kolmogorov System with Time Lags

Abstract

We discuss a general -species discrete Kolmogorov system with time lags. We build some new results about the sufficient conditions for permanence, extinction, and balancing survival. When applying these results to some Lotka-Volterra systems, we obtain the criteria on harmless delay for the permanence as well as profitless delay for balancing survival.

1. Introduction

Difference equations are frequently used in modelling the interactions of populations with nonoverlapping generations (see, e.g., May and Oster [1] for the one-species difference equations; [2] for how populations regulate; Hassel [3], Basson and Fogarty [4] and Beddington et al. [5] for predator-prey models). Sinceone of the most important ecological problems associated with the populations dynamical system is to study the long-term coexistence of all the involved species, such problem in the nondelayed discrete systems had already attracted much attention and thereby many excellent results had appeared ([6–11]). However, recent studies of the natural populations indicated that the interactions of populations, for example, the density-dependent population regulation sometimes takes place over many generations (see [12–29] and the references therein). Turchin [28] evaluated the evidence for delayed density dependence in discrete population dynamics of 14-forest insects, and finded strong evidence for that eight cases exhibited clear evidence for delayed density dependent and lags induce oscillations. And he pointed out that delayed density dependence can arise in natural populations as a result of interactions with other members of the community such as natural enemies, or because high population density may adversely affect the fecundity of the next generation. Furthermore, Turchin and Taylor in [27] proposed the following general delayed discrete populations model:

(1.1)

where and is the density of species at time Recently, Crone [16] showed that inclusion of effects of parental density on offspring mass fundamentally changes population dynamics models by making recruitment a function of population size in two previous generations. Wikan and Mjølhus [30] showed general delay may have different effects on species. By the above conclusions, it is realistic for us to consider the time-delayed discrete population models.

There have been some excellent works devoted to the delayed discrete models ([11, 17, 22, 31–37]). In 1976, Levin and May [31] showed that, similar to the differential-delay equations, those obey nonoverlapping generations with explicit time lags in the density dependent regulatory mechanisms also lead to stable limit cycle behavior. Ginzburg and Taneyhill [17] developed a two-dimensional model of delayed difference equation which relates the average quality of individuals to patterns of abundance. The delayed density dependence was caused by transmission of quality between generations through maternal effects. They proved that the delayed model can produce patterns of population fluctuations. Later, Crone [32] presented a nondelayed model and revealed that the inclusion of delays changes the shape of population cycles (flip versus Hopf bifurcations) and decreases the range of parameters which were used to predict stable equilibria. In [22], Keeling et al. proposed that delayed density dependence should be one of the reasons for what stabilizes the natural enemy-victim interactions and allows the long-term coexistence of the two species. For more mathematical results, Saito et al. [33, 34], Tang and Xiao [35], Kon [11], and Liu et al. [37] proved that time delays are harmless to the coexistence of two-species Lotka-Volterra difference systems. Tang and Xiao [35] studied the two-species Kolmogorov-type delayed discrete system and obtained the sufficient conditions for its permanence. Recently, Liu et al. [37] focused on a general -species discrete competitive Lotka-Volterra system with delayed density dependence and delayed interspecific competition which takes the following form:

(1.2)

Liu et al. [37] showed that under some conditions, the inclusion, exclusion and change of time-delays cannot change the permanence, extinction and balancing survival of species. That is, time-delays maybe harmless for both the permanence and balancing survival of species, in addition to being profitless to the extinction of species. In particular, when , the extinction and permanence of this system were corresponding to some inequalities that only involve the coefficients therein, that is, permanence and extinction in this two-species system are determined only by three elements: growth rate, density dependence and interspecific competition rate.

These papers, while containing many new and significant results, are far from answering the questions on the effects of time delays upon long time behaviors of discrete system. For example, what will be the long-time behaviors for the general nonautonomous delayed discrete Kolmogorov systems? Following the previous works in delay differential models (see [38–46]), in this paper, we study the -species nonautonomous discrete Kolmogorov-type system with time delays, which takes the following form:

(1.3)

where represents the density of population at the th generation; is the space of continuous mapping to with the uniform norm; and is a given function with with some positive below bounds for all , . We define

Suppose is a given integer. We denote with the uniform norm on , that is, for , where is a given norm on . For any function with and any , we define as for , and For the purpose of convenience, we write

In this paper, we assume that system (1.3) always satisfies the following positive initial conditions:

(1.4)

where is continuous. Then we have for all ,

Consequently, we get the general discrete Kolmogorov system (1.3) which embodies both the overlapping interactions among its species and the time-varying environments. Our model extends and joint those models in [11, 33–37].

Definition 1.1.

Species is called permanent if there exists a positive interval such that will ultimately enter and stay in this interval. A population system is called a permanent one (uniformly persistent) if all of its species are permanent.

Definition 1.2.

Species is called extinct if . An -species population system is called -balancing survival () if species in the system go extinct while the remaining being permanent.

The above definitions on permanence and balancing survival in difference systems are equivalent to those usually for differential case (see, e.g., [41–43, 45, 47, 47, 48]).

The purpose of this paper is to construct some general results for the long-time behaviors (permanence and balancing survival) of system (1.3) and study the effects of time delays on the asymptotical behaviors. We get the sufficient conditions for the permanence and balancing survival of system (1.3), which directly extend those in [35]. We also apply the main results for (1.3) to the -species Lotka-Volterra systems of competitive type, which are one of the theoretical interests in population biology since they involve Ricker type (exponential) nonlinearities—one of the standard nonlinearities used in the business. And we obtain the sufficient conditions for system (1.3)'s permanence and balancing survival. These results are applied into the nonautonomous competitive delayed discrete Lotka-Volterra systems and directly generalize some relative results in [33–35, 37]. Moreover, we show the delays do not affect the permanence and balancing survival of the -species Lotka-Volterra discrete systems. Biologically speaking, that is, time delays are both harmless for permanence and profitless to the balancing survival of the system.

Our paper is organized as follows, in the next section we present and prove our main results. In Section 3, we apply the main results into the competitive Lotka-Volterra system and get the corresponding results for its permanence and balancing survival. Discussion follows at the last section.

2. Permanence and Balancing Survival

In ecosystems, the natural resources are limited, so are the species that live in them, therefore, during this paper we always assume that system (1.3) is dissipative, namely system (1.3) is ultimately bounded. Hence there exist a positive constant and positive integer such that for all

Definition 2.1.

A continuous function is said to be a boundary function, if for any , there exist two positive constants such that the following properties hold:

(i)whenever , and , implies that ,

(ii)whenever , and for all imply that for all

Definition 2.2.

We define , a vector Liapunov boundary function of system (1.3), if

(i)there is a boundary function and continuous functions such that for all and

(2.1)

and for all , there is a constant such that

(2.2)

(ii)in addition, for the solution of (1.3) with and , we have

(2.3)

where and there exist some constants , such that whenever

Theorem 2.3.

System (1.3) is permanent if it admits a vector Liapunov boundary function

Remark 2.4.

In [35, Theorem ], Tang and Xiao constructed the sufficient conditions for the permanence of an autonomous two-species Kolmogorov system. Theorem 2.3 directly generalizes their results.

Remark 2.5.

Noting Theorem 2.3 does not need any sign conditions on . Thus we can study several population models simultaneously: competitive, predator-prey, mutual, and so forth.

Using the arguments similar to Lemma in [34], we have the following.

Lemma 2.6.

Assume positive initial conditions hold for system (1.3), then each of its solution is positive with upper and lower bound.

Proof of Theorem 2.3.

We divide the arguments into the following several steps.

Step 1.

Constructing a bounded subset

Suppose is defined as in Definition 2.2 and the vector Liapunov function defined as in Definition 2.2. Let

(2.4)

By Definition 2.2, there exists a positive constant such that for any and , the facts and imply that . Further, one can choose the sufficient small constant with

(2.5)

Define a subset of

(2.6)

Clearly the definition of yields that

In the following we prove that each solution of system (1.3) will eventually enter and stay in , that is, system (1.3) will be permanent.

Step 2.

For any , for and follows that for all

Suppose this false, then there exist some and some such that while and for all unless .

Hence we have two cases to consider.

Case 1 ().

By the definition of , implies . Then by system (1.3),

(2.7)

Using Definition 2.2 we have

Case 2 ().

Since for all , we have . Using the analogous arguments to Case (i), we have . Then we have

(2.8)

Thus we obtain However, since

(2.9)

then we obtain a contradiction, which finishes Step 2.

Step 3.

By Step 2 and the definition that = we obtain that if there exists an integer such that for all , then for all

Step 4.

We claim that there exist and some integer such that for ,

Let be sufficiently large such that for , let , that is, , Following Step 2, we can find a such that for all and By Property (ii) of Definition 2.2, we see that there is a desired such that for Here is dependent on

Using Definition 2.2, we have are bounded for all

Step 5.

We claim that solution enters and stays in for sufficiently large We have two cases to consider.

Case A.

There exists an such that for all ; for this case, Step 3 directly implies the claim.

Case B.

remains in for all large integer , where

(2.10)

In this case, we first claim that there exists a sufficiently large for each such that If it is not true, that is, for all and for each . The definition of implies that for all large , and by the choice of and the definition of , we have , where is some constant with . Hence as A contradiction to the boundedness of (see Step 4). Then for each , there exists a sufficiently large integer such that . By Step 2, for all

Selecting , then we have for all , , that is, for all proving Theorem 2.3.

Definition 2.7.

is called an -boundary function (), if for any and , there exists such that the following properties hold true:

(i)whenever , , and , then

(ii)whenever and for all , then for all

Definition 2.8.

The vector function is called a vector -balancing survival function for system (1.3) if the following properties hold:

(1), where is an -boundary function is a continuous function with such that for and all and any ,

(2.11)

and for all , there exists such that   

(2) for all while for all

Here is a solution of system (1.3) with , admits some , and such that for all , while for sufficiently large , ; as

We have the following theorem.

Theorem 2.9.

Assume there exists a vector -balancing survival Liapunov function for system (1.3), then system (1.3) is -balancing survival, that is, the species are extinct while the rest populations are permanent.

Proof.

First we prove the extinction of species . Let be the vector Liapunov boundary function for system (1.3). By Definition 2.7, for the , we have the following inequality:

(2.12)

then we get which, by Definition 2.8, implies

Now we claim for all and , extinctions of species yield that of species

By Definition 2.8, there exist a positive constant and a sufficiently integer such that for all thus we have

(2.13)

this yields Hence we get the extinctions of species .

Now we prove the permanence of species Choose a with

(2.14)

such that implies that for some if , then for all , where is a constant.

Consider such that for any given , we have; whenever is in the interior of , and , then Such a exists because of property (i) of Definition 2.7. Further by Definition 2.8, we can choose

(2.15)

Let

(2.16)

It is clear that the property (i) of Definition 2.7 yields In the following, we prove that is the desired permanent region for species .

Definition 2.8 and extinctions of species imply that there exist a positive constant and some integer such that for all and

Now we claim that for each , for and imply that for all

For any , for , and follows that for all

Suppose this false, then there exist some and some such that while and for all unless . We have the following two cases to consider.

Case 1 ().

By the definition of , implies . Then by system (1.3),

(2.17)

Using Definition 2.2 we have

Case 2 ().

Since for all , we have . Using the analogous arguments to Case (i), we have . Then we have

(2.18)

Thus we obtain However, we also have

(2.19)

a contradiction, which proves for all

Then using the similararguments to Step 3–5 for Theorem 2.3, we prove Theorem 2.9.

3. Applications to Lotka-Volterra a System

Consider the following nonautonomous discrete competitive systems with time delays:

(3.1)

where represents the density of population at the th generation; is the nonnegative integer delay to the competition between species and species

Denote

(3.2)

for the bounded function with , and let , We assume , and for all , , .

Denote

(3.3)

Then we have the following.

Theorem 3.1.

Assume

() is a nonsymmetric matrix; the vector equation admits a positive solution

()Let be the inverse matrix of with and for all ,

Then system (3.1) is permanent.

Proof.

Using the similar arguments to those in [37], we can prove system (3.1) is dissipative. Let

(3.4)

where are defined in . By Theorem 2.3, we only need to prove the vector function is a vector Liapunov boundary function of system (3.1). With the similar arguments in [45], we can prove that is a boundary function. Hence we only need to prove the (ii) of Definition 2.2.

By (3.4) and system (3.1), we have

(3.5)

By , for all Noticing that is the th column vector of and the th row vector of the matrix , respectively. Then we have

(3.6)

whichfollows

(3.7)

This proves the (ii) of Definition 2.2. Thus we prove is a Liapunov boundary function, proving Theorem 3.1.

Let , Now, we consider the balancing survival of the system (3.1). Denote

(3.8)

Theorem 3.2.

Assume

() is a nonsymmetric matrix and the vector equation admits a positive solution vector with

()Let be the inverse matrix of with and for all and

()For all , there exists such that holds for all

Then system (3.1) is -balancing survival, that is, species are permanent while species will go extinct.

Corollary 3.3.

Assume

()For each , there exists a positive integer with such that holds for all

Then all species in system (3.1) except species 1 are going extinct while species 1 is permanent.

Remark 3.4.

In [37], we considered permanence and balancing survive of system (1.1)—the autonomous case of system (3.1). Theorems 3.1 and 3.2 in this paper generalize the corresponding results in [37].

Remark 3.5.

Kuang [40], Tang and Kuang [45], Liu and Chen [43] obtained the sufficient conditions for the permanence in the delayed -species Lotka-Volterra differential equations. They also proved that time-delays are harmless for the permanence of the continuous Lotka-Volterra system. Our results in Theorem 3.1 and Corollary 3.3 are analogous to theirs.

Remark 3.6.

Theorem 3.1 and Corollary 3.3 can be regarded as the two extreme cases of -balancing survival of system (3.1) with and , respectively. Then Theorem 3.2 unifies Theorem 3.1 and Corollary 3.3.

Remark 3.7.

Noting all conditions in Theorems 3.1 and 3.2 and Corollary 3.3 are independent of the delays , then once conditions for this propositions are satisfied, the inclusion, exclusion or the variations of the time-delays will not affect the conclusions any more.

Proof of Theorem 3.2.

Let with , where

(3.9)

as , and

(3.10)

when

By Theorem 2.9, we only need to prove that is a vector -balancing survival function for system (3.1). With the similar arguments to Theorem in [37], we can prove is an -boundary function for system (3.1); by the dissipative property of system (3.1), we can prove that satisfies conditions for part in Definition 2.8.

For , we have

(3.11)

Thus we have

(3.12)

While for , we have

(3.13)

By , , then Using , we have

(3.14)

which follows

(3.15)

Then also satisfies conditions for part in Definition 2.8, this proves Theorem 3.2.

4. Conclusions

Many authors have studied the effects of time delays on dynamics of population difference systems. Levin and May [31] showed excessive time lags could lead to stable oscillations behaviors. Crone [32] showed that the inclusion of time delays can dramatically change the dynamics and lead to chaos and cyclical. Further, Crone and Taylor [15] proved that inclusion of delays into the density dependence can destabilize the dynamics that may be stabilized by the nondelayed density dependence. Ginzburg and Taneybill [17] obtained that delays can produce patterns of population fluctuation. Keeling et al. [22, 45] showed that time delays might be one of the causes to stabilize the natural enemy victim interactions and allow the long term coexistence of the two species.

Harmless delays have been well-known for some continuous population since Wang and Ma [46] proved that delays are "harmless" for the permanence of a continuous Lotka-Volterra predator-prey system, similar conclusions can also be found in some competitive Lotka-Volterra systems (see [43, 49, 50]). Recently, Liu and Chen [43] proved the existence of "profitless delays", that is, the delays do not affect on species' extinction. For the discrete system, to study the effects of time delays on permanence, Tang and Xiao [35], Saito et al. [33] and Liu et al. [37] study the effects of time delays on the two-species competitive systems and they prove that time delays are "harmless" for the uniform persistence or permanence. Saito et al. [34] also discover the same conclusions for the two-species predator-prey systems.

Different from the above results, we consider the long-time behaviors of the discrete nonautonomous Kolmogorov-type population system with delays. We obtained the sufficient conditions for its permanence and balancing survival behaviors. These results have the advantage that we do not assume any sign condition on . So, we can study simultaneously several population models: competing species, predator-prey, mutualism, and so forth. In this paper, we have only applied the main results to Lotka-Volterra competing species.

When applying the results of Kolmogorov system into the nonautonomous the competitive system of Lotka-Volterra type, we construct the sufficient conditions for the permanence and balancing survival behaviors of these systems, with all the conditions independent of the time-delays. Hence if the nondelayed system is permanent, its corresponding delayed system will be permanent, too. If several species of the nondelayed systems are balancing survival, so will be in the corresponding delayed system. On the other hand, under the corresponding conditions, if the delayed system is permanent or some of its species go extinct (balancing survival), so will be in the relative nondelayed system.

Thus, under the proper conditions, neither can time delays break the permanence of some species into extinction, nor can they save the extinction of some species. Therefore, time-lags in the discrete competitive Lotka-Volterra system with time-varying environments are both harmless for the permanence and profitless to the extinction of species in system (3.1), these results confirm and improve our previous conclusions for the discrete autonomous Lotka-Volterra systems [37].

Further, we show that the permanence and extinction of the discrete system (3.1) are equivalent to their corresponding continuous systems (see [40, 43, 45]), where time delays are also both harmless for the permanence and profitless to the extinction of species of the system.

Time delays have been shown to dramatically change the dynamics of the discrete populations systems (see [15, 17, 22, 32]) and they may even lead to some complicated dynamical behaviors such as Crone [32]. Based on our results, it would be interesting to consider the effects of time delay on the stability of discrete systems, we leave this as our future work.