Permanence and universal classification of discrete-time competitive systems via the carrying simplex

Mats Gyllenberg; Jifa Jiang; Lei Niu; Ping Yan

doi:10.3934/dcds.2020088

Article Contents

2020, Volume 40, Issue 3: 1621-1663. Doi: 10.3934/dcds.2020088

This issue Previous Article Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrödinger system Next Article On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay

Permanence and universal classification of discrete-time competitive systems via the carrying simplex

1.
Department of Mathematics and Statistics, University of Helsinki, Helsinki FI-00014, Finland
2.
Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China
3.
School of Sciences, Zhejiang A & F University, Hangzhou 311300, China

^* Corresponding author: Lei Niu
^* Corresponding author: Lei Niu

Received: March 31, 2019

Revised: August 31, 2019

Published: December 2019

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11371252 and Grant No. 11771295, Shanghai Gaofeng Project for University Academic Program Development, and the Academy of Finland

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Abstract

We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of $ 33 $ stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes $ 1-25 $ and $ 33 $; there is always a heteroclinic cycle in class $ 27 $; Neimark-Sacker bifurcations may occur in classes $ 26-31 $ but cannot occur in class $ 32 $. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes $ 29, 31, 33 $ and those in class $ 27 $ with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.

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Mathematics Subject Classification: Primary: 37B25, 37Cxx, 37N25; Secondary: 92D25.

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Figure 1. A carrying simplex $ \Sigma $ with a repelling heteroclinic cycle $ \partial\Sigma $

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Figure 2. The phase portrait on $ \Sigma $ replaced by $ \Delta^1 $. A closed dot $ \bullet $ denotes a fixed point which attracts on $ \Sigma $, and an open dot $ \circ $ denotes the one which repels on $ \Sigma $. Each $ \Sigma $ stands for an equivalence class. Class $ 1 $ corresponds to Proposition 4.7 (a) and (b); class $ 2 $ corresponds to Proposition 4.7 (c); class $ 3 $ corresponds to Proposition 4.7 (d)

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Figure 3. The phase portrait on $ \Sigma $ for class $ 33 $. Every orbit in the interior of $ \Sigma $ converges to $ p $. The fixed point notation is as in Table 1

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Figure 4. The phase portrait on $ \Sigma $ for class $ 29 $. The fixed point notation is as in Table 1

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Figure 5. The orbit emanating from $ x_0 = (1, 0.0667, 0.0667) $ for the map $ T\in\mathrm{CLG}(3) $ with the parameter matrix $ U $ given in Example 5.1 and $ r_1 = 1, r_2 = 0.2, r_3 = 1 $ leads away from $ \partial \Sigma $ and tends to an attracting invariant closed curve, and the orbit emanating from $ x_0 = (0.2151, 0.746, 0.0173) $ also tends to an attracting invariant closed curve

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Figure 6. The orbit emanating from $ x_0 = (1, 0.0667, 0.0667) $ for the map $ T\in\mathrm{CGAA}(3) $ with the parameter matrix $ U $ given in Example 5.1 and $ r_1 = r_2 = r_3 = 1 $, $ c_1 = \frac{1}{10}, c_2 = \frac{1}{5}, c_3 = \frac{1}{5} $ leads away from $ \partial \Sigma $ and tends to an attracting invariant closed curve, and the orbit emanating from $ x_0 = (0.7, 0.1642, 0.1685) $ also tends to an attracting invariant closed curve

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Figure 7. The orbit emanating from $ x_0 = (0.04, 0.12, 0.36) $ for the map $ T\in\mathrm{CGAA}(3) $ with the parameter matrix $ U $ given in Example 5.3 and $ r_1 = r_2 = r_3 = 1 $, $ c_1 = 0.1, c_2 = 0.79, c_3 = 0.1 $ tends to an attracting invariant closed curve, while the orbit emanating from $ x_0 = (0.0002, 0.023, 0.486) $ approaches the heteroclinic cycle $ \partial \Sigma $

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Figure 8. The orbit emanating from $ x_0 = (0.427, 0.8574, 0.014) $ for the map $ T\in\mathrm{MFC}(3) $ with the parameter matrix $ U $ given in Example 5.4, $ c = \frac{4}{5} $ and $ r_1 = r_3 = 1, r_2 = 0.03 $ tends to an attracting invariant closed curve

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Figure 9. The orbit emanating from $ x_0 = (0.5962, 0.4857, 0.193) $ for the map $ T\in\mathrm{MFC}(3) $ with the parameter matrix $ U $ given in Example 5.5, $ c = \frac{4}{5} $ and $ r_1 = r_3 = 1, r_2 = 0.02 $ tends to an attracting invariant closed curve

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Figure 10. The orbit emanating from $ x_0 = (0.3128, 0.8347, 0.0199) $ for the map $ T\in\mathrm{CRC}(3) $ with the parameter matrix $ U $ given in Example 5.4 and $ r_1 = \frac{1}{11}, r_2 = 0.01, r_3 = \frac{2}{7} $ tends to an attracting invariant closed curve

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Table 1. The $33$ equivalence classes in $\mathrm{DCS}(3, f)$, where $\gamma_{ij} = \mu_{ii}-\mu_{ji}$, $\beta_{ij} = \frac{\mu_{jj}-\mu_{ij}}{\mu_{ii}\mu_{jj}-\mu_{ij}\mu_{ji}}$ ($\beta_{ij}$ is well defined; see Remark 4.6), $i, j = 1, 2, 3$ and $i\neq j$, and each $\Sigma$ is given by a representative map of that class. A fixed point is represented by a closed dot $\bullet$ if it attracts on $\Sigma$, by an open dot $\circ$ if it repels on $\Sigma$, and by the intersection of its stable and unstable manifolds if it is a saddle on $\Sigma$. For classes $1-25$ and $33$, every orbit converges to some fixed point; for classes $26-31$, Neimark-Sacker bifurcations might occur; for class $27$, $\partial \Sigma$ is a heteroclinic cycle; for class $32$, the unique positive fixed point is a repeller and Neimark-Sacker bifurcation cannot occur in this class

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[1]	L. J. S. Allen, E. J. Allen and D. N. Atkinson, Integrodifference equations applied to plant dispersal, competition, and control, in Differential Equations with Applications to Biology, Fields Institute Communications, 21, Amer. Math. Soc., Providence, RI, 1999, 15–30.
[2]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114.
[3]	D. N. Atkinson, Mathematical Models for Plant Competition and Dispersal, Master's thesis, Texas Tech University in Lubbock, 1997.
[4]	S. Baigent, Geometry of carrying simplices of 3-species competitive Lotka-Volterra systems, Nonlinearity, 26 (2013), 1001-1029. doi: 10.1088/0951-7715/26/4/1001.
[5]	S. Baigent, Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems, J. Difference Equ. Appl., 22 (2016), 609-622. doi: 10.1080/10236198.2015.1125895.
[6]	S. Baigent, Convex geometry of the carrying simplex for the May–Leonard map, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1697-1723. doi: 10.3934/dcdsb.2018288.
[7]	S. Baigent and Z. Hou, Global stability of interior and boundary fixed points for Lotka-Volterra systems, Differ. Equ. Dyn. Syst., 20 (2012), 53-66. doi: 10.1007/s12591-012-0103-0.
[8]	S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, J. Difference Equ. Appl., 23 (2017), 1378-1396. doi: 10.1080/10236198.2017.1333116.
[9]	E. C. Balreira, S. Elaydi and R. Luís, Global stability of higher dimensional monotone maps, J. Difference Equ. Appl., 23 (2017), 2037-2071. doi: 10.1080/10236198.2017.1388375.
[10]	Å. Brännström and D. J. T. Sumpter, The role of competition and clustering in population dynamics, Proc. R. Soc. B, 272 (2005), 2065-2072. doi: 10.1098/rspb.2005.3185.
[11]	X. Chen, J. Jiang and L. Niu, On Lotka-Volterra equations with identical minimal intrinsic growth rate, SIAM J. Appl. Dyn. Syst., 14 (2015), 1558-1599. doi: 10.1137/15M1006878.
[12]	S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Fundamental Principles of Mathematical Science, 251, Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4.
[13]	J. M. Cushing, On the fundamental bifurcation theorem for semelparous Leslie models, in Dynamics, Games and Science, CIM Ser. Math. Sci., 1, Springer, Cham, 2015, 215–251.
[14]	J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Difference Equ. Appl., 10 (2004), 1139-1151. doi: 10.1080/10236190410001652739.
[15]	N. V. Davydova, O. Diekmann and S. A. van Gils, On circulant populations. Ⅰ. The algebra of semelparity, Linear Algebra Appl., 398 (2005), 185-243. doi: 10.1016/j.laa.2004.12.020.
[16]	P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol., 11 (1981), 319-335. doi: 10.1007/BF00276900.
[17]	O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52. doi: 10.3934/dcds.2008.20.37.
[18]	H. T. M. Eskola and S. A. H. Geritz, On the mechanistic derivation of various discrete-time population models, Bull. Math. Biol., 69 (2007), 329-346. doi: 10.1007/s11538-006-9126-4.
[19]	M. A. Fishman, Density effects in population growth: An exploration, Biosystems, 40 (1997), 219-236. doi: 10.1016/S0303-2647(96)01649-8.
[20]	J. E. Franke and A.-A. Yakubu, Mutual exclusion versus coexistence for discrete competitive systems, J. Math. Biol., 30 (1991), 161-168. doi: 10.1007/BF00160333.
[21]	J. E. Franke and A.-A. Yakubu, Geometry of exclusion principles in discrete systems, J. Math. Anal. Appl., 168 (1992), 385-400. doi: 10.1016/0022-247X(92)90167-C.
[22]	B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations, SIAM J. Math. Anal., 34 (2003), 1007-1039. doi: 10.1137/S0036141001392815.
[23]	S. A. H. Geritz, Resident-invader dynamics and the coexistence of similar strategies, J. Math. Biol., 50 (2005), 67-82. doi: 10.1007/s00285-004-0280-8.
[24]	S. A. H. Geritz, M. Gyllenberg, F. J. A. Jacobs and K. Parvinen, Invasion dynamics and attractor inheritance, J. Math. Biol., 44 (2002), 548-560. doi: 10.1007/s002850100136.
[25]	S. A. H. Geritz and E. Kisdi, On the mechanistic underpinning of discrete-time population models with complex dynamics, J. Theoret. Biol., 228 (2004), 261-269. doi: 10.1016/j.jtbi.2004.01.003.
[26]	S. A. H. Geritz, E. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evolutionary Ecology, 12 (1998), 35-57. doi: 10.1023/A:1006554906681.
[27]	S. A. H. Geritz, J. A. J. Metz, E. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching, Phys. Rev. Lett., 78 (1997), 2024-2027. doi: 10.1103/PhysRevLett.78.2024.
[28]	W. Govaerts, R. K. Ghaziani, Y. A. Kuznetsov and H. G. E. Meijer, Numerical methods for two-parameter local bifurcation analysis of maps, SIAM J. Sci. Comput., 29 (2007), 2644-2667. doi: 10.1137/060653858.
[29]	W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and N. Neirynck, A study of resonance tongues near a Chenciner bifurcation using MatcontM, European Nonlinear Dynamics Conference, 2011, 24–29.
[30]	A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.
[31]	M. Gyllenberg, I. Hanski and T. Lindström, Continuous versus discrete single species population models with adjustable reproductive strategies, Bull. Math. Biol., 59 (1997), 679-705. doi: 10.1007/BF02458425.
[32]	M. Gyllenberg, J. Jiang and L. Niu, A note on global stability of three-dimensional Ricker models, J. Difference Equ. Appl., 25 (2019), 142-150. doi: 10.1080/10236198.2019.1566459.
[33]	M. Gyllenberg, J. Jiang, L. Niu and P. Yan, On the dynamics of multi-species Ricker models admitting a carrying simplex, J. Difference Equ. Appl., in press. doi: 10.1080/10236198.2019.1663182.
[34]	M. Gyllenberg, J. Jiang, L. Niu and P. Yan, On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex, Discrete Contin. Dyn. Syst., 38 (2018), 615-650. doi: 10.3934/dcds.2018027.
[35]	M. Gyllenberg, P. Yan and Y. Wang, A 3D competitive Lotka-Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So, Appl. Math. Lett., 19 (2006), 1-7. doi: 10.1016/j.aml.2005.01.002.
[36]	J. K. Hale and A. S. Somolinos, Competition for fluctuating nutrient, J. Math. Biol., 18 (1983), 255-280. doi: 10.1007/BF00276091.
[37]	M. P. Hassell, Density-dependence in single-species populations, J. Anim. Ecol., 44 (1975), 283-295. doi: 10.2307/3863.
[38]	M. P. Hassell and H. N. Comins, Discrete time models for two-species competition, Theoret. Population Biology, 9 (1976), 202-221. doi: 10.1016/0040-5809(76)90045-9.
[39]	M. W. Hirsch, Systems of differential equations which are competitive or cooperative. Ⅲ. Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003.
[40]	M. W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn., 2 (2008), 169-179. doi: 10.1080/17513750801939236.
[41]	J. Hofbauer, Heteroclinic cycles in ecological differential equations, Tatra Mt. Math. Publ., 4 (1994), 105-116.
[42]	J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol., 25 (1987), 553-570. doi: 10.1007/BF00276199.
[43]	J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.
[44]	J. Hofbauer and J. W.-H. So, Multiple limit cycles for three dimensional Lotka-Volterra equations, Appl. Math. Lett., 7 (1994), 65-70. doi: 10.1016/0893-9659(94)90095-7.
[45]	Z. Hou and S. Baigent, Global stability and repulsion in autonomous Kolmogorov systems, Commun. Pure Appl. Anal., 14 (2015), 1205-1238. doi: 10.3934/cpaa.2015.14.1205.
[46]	T. Hüls and C. Pötzsche, Qualitative analysis of a nonautonomous Beverton-Holt Ricker model, SIAM J. Appl. Dyn. Syst., 13 (2014), 1442-1488. doi: 10.1137/140955434.
[47]	V. Hutson and W. Moran, Persistence of species obeying difference equations, J. Math. Biol., 15 (1982), 203-213. doi: 10.1007/BF00275073.
[48]	J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36 (2016), 217-244. doi: 10.3934/dcds.2016.36.217.
[49]	J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74 (2017), 1223-1261. doi: 10.1007/s00285-016-1052-y.
[50]	J. Jiang, L. Niu and Y. Wang, On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol., 72 (2016), 939-972. doi: 10.1007/s00285-015-0920-1.
[51]	J. Jiang, L. Niu and D. Zhu, On the complete classification of nullcline stable competitive three-dimensional Gompertz models, Nonlinear Anal. Real World Appl., 20 (2014), 21-35. doi: 10.1016/j.nonrwa.2014.04.006.
[52]	F. G. W. Jones and J. N. Perry, Modelling populations of cyst-nematodes (Nematoda: Heteroderidae), J. Applied Ecology, 15 (1978), 349-371. doi: 10.2307/2402596.
[53]	R. Kon, Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points, J. Math. Biol., 48 (2004), 57-81. doi: 10.1007/s00285-003-0224-8.
[54]	R. Kon, Convex dominates concave: An exclusion principle in discrete-time Kolmogorov systems, Proc. Amer. Math. Soc., 134 (2006), 3025-3034. doi: 10.1090/S0002-9939-06-08309-2.
[55]	R. Kon and Y. Takeuchi, Permanence of host-parasitoid systems, Nonlinear Anal., 47 (2001), 1383-1393. doi: 10.1016/S0362-546X(01)00273-5.
[56]	Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4757-3978-7.
[57]	Y. A. Kuznetsov and R. J. Sacker, Neimark-Sacker bifurcation, Scholarpedia, 3 (2008). doi: 10.4249/scholarpedia.1845.
[58]	R. Law and A. R. Watkinson, Response-surface analysis of two-species competition: An experiment on Phleum arenarium and Vulpia fasciculata, J. Ecol., 75 (1987), 871-886. doi: 10.2307/2260211.
[59]	P. H. Leslie and J. C. Gower, The properties of a stochastic model for two competing species, Biometrika, 45 (1958), 316-330. doi: 10.1093/biomet/45.3-4.316.
[60]	J. M. Levine and M. Rees, Coexistence and relative abundance in annual plant assemblages: The roles of competition and colonization, Amer. Naturalist, 160 (2002), 452-467. doi: 10.1086/342073.
[61]	Z. Lu and Y. Luo, Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle, Comput. Math. Appl., 46 (2003), 231-238. doi: 10.1016/S0898-1221(03)90027-7.
[62]	Z. Lu and W. Wang, Permanence and global attractivity for Lotka-Volterra difference systems, J. Math. Biol., 39 (1999), 269-282. doi: 10.1007/s002850050171.
[63]	R. M. May, Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos, Science, 186 (1974), 645-647. doi: 10.1126/science.186.4164.645.
[64]	R. M. May and G. F. Oster, Bifurcations and dynamic complexity in simple ecological models, Amer. Naturalist, 110 (1976), 573-599. doi: 10.1086/283092.
[65]	C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719512.
[66]	J. Mierczyński, The ${C}^1$ property of convex carrying simplices for competitive maps, Ergodic Theory Dynam. Systems, (2018), 1–16. doi: 10.1017/etds.2018.85.
[67]	J. Mierczyński, The ${C}^1$ property of convex carrying simplices for three-dimensional competitive maps, J. Difference Equ. Appl., 24 (2018), 1199-1209. doi: 10.1080/10236198.2018.1428964.
[68]	J. Mierczyński, L. Niu and A. Ruiz-Herrera, Linearization and invariant manifolds on the carrying simplex for competitive maps, J. Differential Equations, 267 (2019), 7385-7410. doi: 10.1016/j.jde.2019.08.001.
[69]	L. Niu and A. Ruiz-Herrera, Trivial dynamics in discrete-time systems: Carrying simplex and translation arcs, Nonlinearity, 31 (2018), 2633-2650. doi: 10.1088/1361-6544/aab46e.
[70]	M. Rees and M. Westoby, Game-theoretical evolution of seed mass in multi-species ecological models, Oikos, 78 (1997), 116-126. doi: 10.2307/3545807.
[71]	W. E. Ricker, Stock and recruitment, J. Fish. Res. Board. Can., 11 (1954), 559-623. doi: 10.1139/f54-039.
[72]	L.-I. W. Roeger, Discrete May-Leonard competition models. Ⅱ, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 841-860. doi: 10.3934/dcdsb.2005.5.841.
[73]	L.-I. W. Roeger and L. J. S. Allen, Discrete May–Leonard competition models. Ⅰ, J. Difference Equ. Appl., 10 (2004), 77-98. doi: 10.1080/10236190310001603662.
[74]	A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Difference Equ. Appl., 19 (2013), 96-113. doi: 10.1080/10236198.2011.628663.
[75]	H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194. doi: 10.1016/0022-0396(86)90086-0.
[76]	H. L. Smith, Planar competitive and cooperative difference equations, J. Differ. Equations Appl., 3 (1998), 335-357. doi: 10.1080/10236199708808108.
[77]	H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, American Mathematical Society, Providence, RI, 2011.
[78]	C. R. Townsend, M. Begon and J. L. Harper, Essentials of Ecology, Blackwell Publishing, 2008.
[79]	W. Van den berg, W. A. H. Rossing and J. Grasman, Contest and scramble competition and the carry-over effect in Globodera spp. in potato-based crop rotations using an extended Ricker model, J. Nematol., 38 (2006), 210-220.
[80]	P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka–Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. doi: 10.1137/S0036139995294767.
[81]	G. C. Varley, G. R. Gradwell and M. P. Hassell, Insect Population Ecology, Blackwell Scientific Publications, Oxford, 1973.
[82]	Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6.
[83]	D. Xiao and W. Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Differential Equations, 164 (2000), 1-15. doi: 10.1006/jdeq.1999.3729.
[84]	E. C. Zeeman and M. L. Zeeman, On the convexity of carrying simplices in competitive Lotka-Volterra systems, in Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, 1994, 353–364.
[85]	E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Amer. Math. Soc., 355 (2003), 713-734. doi: 10.1090/S0002-9947-02-03103-3.
[86]	E. C. Zeeman and M. L. Zeeman, An $n$-dimensional competitive Lotka-Volterra system is generically determined by the edges of its carrying simplex, Nonlinearity, 15 (2002), 2019-2032. doi: 10.1088/0951-7715/15/6/312.
[87]	M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217. doi: 10.1080/02681119308806158.