# Asymptotical Behaviors of Nonautonomous Discrete Kolmogorov System with Time Lags

- Shengqiang Liu
^{1}Email author

**2010**:517378

https://doi.org/10.1155/2010/517378

© Shengqiang Liu. 2010

**Received: **23 November 2009

**Accepted: **24 April 2010

**Published: **31 May 2010

## 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.

## Keywords

## 1. Introduction

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.

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.

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

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

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.

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.

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.

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:

(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:

and for all , there exists such that

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.

then we get which, by Definition 2.8, implies

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

this yields Hence we get the extinctions of species .

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

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.

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

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

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

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.

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.

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

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.

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.

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.

## Declarations

### Acknowledgments

The authors are grateful to the anonymous referees for their careful reading and valuable comments, which led to an improvement of their original manuscript. This work is supported by the National Natural Science Foundation of China (no.10601042), Science Research Foundation in Harbin Institute of Technology (HITC200714) and the program of excellent Team in Harbin Institute of Technology. The authors would like to thank Dr. S. Tang and Prof. M. J. Keeling for sending their reprints/preprints to them.

## Authors’ Affiliations

## References

- May RM, Oster GF:
**Bifurcations and dynamic complexity in simple ecological models.***The American Naturalist*1976,**110**(974):573-599. 10.1086/283092View ArticleGoogle Scholar - May RM:
**The croonian lecture, 1985. When two and two do not make four: nonlinear phenomena in ecology.***Proceedings of the Royal Society of London*1986,**228**(1252):241-266. 10.1098/rspb.1986.0054View ArticleGoogle Scholar - Hassell MP:
*The Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology*.*Volume 13*. Princeton University Press, Princeton, NJ, USA; 1978:vii+237.Google Scholar - Basson M, Fogarty MJ:
**Harvesting in discrete-time predator-prey systems.***Mathematical Biosciences*1997,**141**(1):41-74. 10.1016/S0025-5564(96)00173-3MATHView ArticleGoogle Scholar - Beddington JR, Free CA, Lawton JH:
**Dynamic complexity in predator prey models framed in difference equations.***Nature*1975,**255**(5503):58-60. 10.1038/255058a0View ArticleGoogle Scholar - Agarwal RP:
*Difference Equations and Inequalities: Theorey, Methods and Applications, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 155*. Marcel Dekker, New York, NY, USA; 1992:xiv+777.Google Scholar - Chan DM, Franke JE:
**Extinction, weak extinction and persistence in a discrete, competitive Lotka-Volterra model.***International Journal of Applied Mathematics and Computer Science*2000,**10**(1):7-36.MATHMathSciNetGoogle Scholar - Hofbauer J, Hutson V, Jansen W:
**Coexistence for systems governed by difference equations of Lotka-Volterra type.***Journal of Mathematical Biology*1987,**25**(5):553-570. 10.1007/BF00276199MATHMathSciNetView ArticleGoogle Scholar - Hutson V, Moran W:
**Persistence of species obeying difference equations.***Journal of Mathematical Biology*1982,**15**(2):203-213. 10.1007/BF00275073MATHMathSciNetView ArticleGoogle Scholar - Lu Z, Wang W:
**Permanence and global attractivity for Lotka-Volterra difference systems.***Journal of Mathematical Biology*1999,**39**(3):269-282. 10.1007/s002850050171MATHMathSciNetView ArticleGoogle Scholar - Kon R:
**Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points.***Journal of Mathematical Biology*2004,**48**(1):57-81. 10.1007/s00285-003-0224-8MATHMathSciNetView ArticleGoogle Scholar - Arditi R, Ginzburg LR:
**Coupling in predator-prey dynamics: ratio-dependence.***Journal of Theoretical Biology*1989,**139**(3):311-326. 10.1016/S0022-5193(89)80211-5View ArticleGoogle Scholar - Chitty D:
**Population proceses in the vole and their relevance to general theory.***Canadian Journal of Zoology*1960,**38:**99-113. 10.1139/z60-011View ArticleGoogle Scholar - Chitty D:
*Do Lemmings Commit Suicide? Beautiful Hypotheses and Ugly Facts*. Oxford University Press, New York, NY, USA; 1996.Google Scholar - Crone EE, Taylor DR:
**complex dynamics in experimental populations of an annual plant, Cardamine Pensylvanica.***Ecology*1996,**77**(1):289-299. 10.2307/2265678View ArticleGoogle Scholar - Crone EE:
**Parental environmental effects and cyclical dynamics in plant populations.***The American Naturalist*1997,**150**(6):708-729. 10.1086/286090View ArticleGoogle Scholar - Ginzburg LR, Taneyhill DE:
**Population cycles of forest Lepidoptera: a maternal effect hypothesis.***Journal of Animal Ecology*1994,**63**(1):79-92. 10.2307/5585View ArticleGoogle Scholar - Ginzburg LR:
**Assuming reproduction to be a function of consumption raises doubts about some popular predator-prey models.***Journal of Animal Ecology*1998,**67**(2):325-327. 10.1046/j.1365-2656.1998.00226.xView ArticleGoogle Scholar - Holyoak M:
**Identifying delayed density dependence in time-series data.***Oikos*1994,**70**(2):296-304. 10.2307/3545641View ArticleGoogle Scholar - Hornfeldt B:
**Delayed density dependence as a determinant of vole cycles.***Ecology*1994,**75**(3):791-806. 10.2307/1941735View ArticleGoogle Scholar - Inchausti P, Ginzburg LR:
**Small mammals cycles in northern Europe: patterns and evidence for a maternal effect hypothesis.***Journal of Animal Ecology*1998,**67**(2):180-194. 10.1046/j.1365-2656.1998.00189.xView ArticleGoogle Scholar - Keeling MJ, Wilson HB, Pacala SW:
**Reinterpreting space, time lags, and functional responses in ecological models.***Science*2000,**290**(5497):1758-1761. 10.1126/science.290.5497.1758View ArticleGoogle Scholar - Mousseau TA, Dingle H:
**Maternal effects in insect life histories.***Annual Review of Entomology*1991,**36**(1):511-534. 10.1146/annurev.en.36.010191.002455View ArticleGoogle Scholar - Mousseau T, Fox CW:
*Maternal Effects as Adaptations*. Oxford University Press, New York, NY, USA; 1998.Google Scholar - Roach DA, Wulff RD:
**Maternal effects in plants.***Annual Review of Ecology and Systematics*1987,**18:**209-235. 10.1146/annurev.es.18.110187.001233View ArticleGoogle Scholar - Rossiter MC:
**Maternal effects hypothesis of herbivore outbreak.***BioScience*1994,**44**(11):752-763. 10.2307/1312584View ArticleGoogle Scholar - Turchin P, Taylor AD:
**Complex dynamics in ecological time series.***Ecology*1992,**73**(1):289-305. 10.2307/1938740View ArticleGoogle Scholar - Turchin P:
**Rarity of density dependence or population regulation with lags?***Nature*1990,**344**(6267):660-663. 10.1038/344660a0View ArticleGoogle Scholar - Wikan A:
**From chaos to chaos. An analysis of a discrete age-structured prey-predator model.***Journal of Mathematical Biology*2001,**43**(6):471-500. 10.1007/s002850100101MATHMathSciNetView ArticleGoogle Scholar - Wikan A, Mjølhus E:
**Overcompensatory recruitment and generation delay in discrete age-structured population models.***Journal of Mathematical Biology*1996,**35**(2):195-239. 10.1007/s002850050050MATHMathSciNetView ArticleGoogle Scholar - Levin SA, May RM:
**A note on difference-delay equations.***Theoretical Population Biology*1976,**9**(2):178-187. 10.1016/0040-5809(76)90043-5MATHMathSciNetView ArticleGoogle Scholar - Crone EE:
**Delayed density dependence and the stability of interacting populations and subpopulations.***Theoretical Population Biology*1997,**51**(1):67-76. 10.1006/tpbi.1997.1309MATHView ArticleGoogle Scholar - Saito Y, Ma W, Hara T:
**A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays.***Journal of Mathematical Analysis and Applications*2001,**256**(1):162-174. 10.1006/jmaa.2000.7303MATHMathSciNetView ArticleGoogle Scholar - Saito Y, Hara T, Ma W:
**Harmless delays for permanence and impersistence of a Lotka-Volterra discrete predator-prey system.***Nonlinear Analysis: Theory, Methods & Applications*2002,**50**(5):703-715. 10.1016/S0362-546X(01)00778-7MATHMathSciNetView ArticleGoogle Scholar - Tang S, Xiao Y:
**Permanence in Kolmogorov-type systems of delay difference equations.***Journal of Difference Equations and Applications*2001,**7**(2):167-181. 10.1080/10236190108808267MATHMathSciNetView ArticleGoogle Scholar - Wendi W, Mulone G, Salemi F, Salone V:
**Global stability of discrete population models with time delays and fluctuating environment.***Journal of Mathematical Analysis and Applications*2001,**264**(1):147-167. 10.1006/jmaa.2001.7666MATHMathSciNetView ArticleGoogle Scholar - Liu S, Chen L, Agarwal RP:
**Harmless and profitless delays in discrete competitive Lotka-Volterra systems.***Applicable Analysis*2004,**83**(4):411-431. 10.1080/00036810310001643202MATHMathSciNetView ArticleGoogle Scholar - Freedman HI, Ruan SG:
**Uniform persistence in functional-differential equations.***Journal of Differential Equations*1995,**115**(1):173-192. 10.1006/jdeq.1995.1011MATHMathSciNetView ArticleGoogle Scholar - Freedman HI, Agarwal M, Devi S:
**Analysis of stability and persistence in a ratio-dependent predator-prey resource model.***International Journal of Biomathematics*2009,**2**(1):107-118. 10.1142/S1793524509000522MathSciNetView ArticleGoogle Scholar - Kuang Y:
*Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering*.*Volume 191*. Academic Press, Boston, Mass, USA; 1993:xii+398.Google Scholar - Liu S, Chen L:
**Necessary-sufficient conditions for permanence and extinction in Lotka-Volterra system with distributed delays.***Applied Mathematics Letters*2003,**16**(6):911-917. 10.1016/S0893-9659(03)90016-4MATHMathSciNetView ArticleGoogle Scholar - Liu S, Chen L:
**Necessary-sufficient conditions for permanence and extinction in Lotka-Volterra system with discrete delays.***Applicable Analysis*2002,**81**(3):575-587. 10.1080/0003681021000004302MATHMathSciNetView ArticleGoogle Scholar - Liu S, Chen L:
**Permanence, extinction and balancing survival in nonautonomous Lotka-Volterra system with delays.***Applied Mathematics and Computation*2002,**129**(2-3):481-499. 10.1016/S0096-3003(01)00058-3MATHMathSciNetView ArticleGoogle Scholar - Nakata Y:
**Permanence for the Lotka-Volterra cooperative system with several delays.***International Journal of Biomathematics*2009,**2**(3):267-285. 10.1142/S1793524509000716MathSciNetView ArticleGoogle Scholar - Tang B, Kuang Y:
**Permanence in Kolmogorov-type systems of nonautonomous functional-differential equations.***Journal of Mathematical Analysis and Applications*1996,**197**(2):427-447. 10.1006/jmaa.1996.0030MATHMathSciNetView ArticleGoogle Scholar - Wendi W, Zhien M:
**Harmless delays for uniform persistence.***Journal of Mathematical Analysis and Applications*1991,**158**(1):256-268. 10.1016/0022-247X(91)90281-4MATHMathSciNetView ArticleGoogle Scholar - Montes de Oca F, Zeeman ML:
**Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems.***Journal of Mathematical Analysis and Applications*1995,**192**(2):360-370. 10.1006/jmaa.1995.1177MATHMathSciNetView ArticleGoogle Scholar - Ahmad S, Montes de Oca F:
**Extinction in nonautonomous****-periodic competitive Lotka-Volterra system.***Applied Mathematics and Computation*1998,**90**(2-3):155-166. 10.1016/S0096-3003(97)00396-2MATHMathSciNetView ArticleGoogle Scholar - Lu ZY, Takeuchi Y:
**Permanence and global attractivity for competitive Lotka-Volterra systems with delay.***Nonlinear Analysis: Theory, Methods & Applications*1994,**22**(7):847-856. 10.1016/0362-546X(94)90053-1MATHMathSciNetView ArticleGoogle Scholar - Fujimoto H:
**Dynamical behaviors for population growth equations with delays.***Nonlinear Analysis: Theory, Methods & Applications*1998,**31**(5-6):549-558. 10.1016/S0362-546X(97)00421-5MATHMathSciNetView ArticleGoogle Scholar

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