1555:
large regions with clear boundaries. As predicted by the theory, chaos was also found; taking place however over much smaller islands of the parameter space which causes difficulties in the identification of their location by a random search algorithm. These regions where chaos occurs are, in the three cases analyzed in, situated at the interface between a non-chaotic four species region and a region where extinction occurs. This implies a high sensitivity of biodiversity with respect to parameter variations in the chaotic regions. Additionally, in regions where extinction occurs which are adjacent to chaotic regions, the computation of local
Lyapunov exponents revealed that a possible cause of extinction is the overly strong fluctuations in
2821:
1568:
2404:
2900:
1937:
3160:
1102:
472:
1674:
2816:{\displaystyle {\begin{aligned}\operatorname {Re} (\lambda _{k})&=\operatorname {Re} \left(1+\alpha _{-2}e^{i2\pi k(N-2)/N}+\alpha _{-1}e^{i2\pi k(N-1)/N}+\alpha _{1}e^{i2\pi k/N}+\alpha _{2}e^{i4\pi k/N}\right)\\&=1+(\alpha _{-2}+\alpha _{2})\cos \left({\frac {4\pi k}{N}}\right)+(\alpha _{-1}+\alpha _{1})\cos \left({\frac {2\pi k}{N}}\right)>0\end{aligned}}}
1296:
2917:
191:
1131:
1523:
1608:
There are many situations where the strength of species' interactions depends on the physical distance of separation. Imagine bee colonies in a field. They will compete for food strongly with the colonies located near to them, weakly with further colonies, and not at all with colonies that are far
1932:{\displaystyle \alpha _{ij}={\begin{bmatrix}1&\alpha _{1}&0&0&\alpha _{-1}\\\alpha _{-1}&1&\alpha _{1}&0&0\\0&\alpha _{-1}&1&\alpha _{1}&0\\0&0&\alpha _{-1}&1&\alpha _{1}\\\alpha _{1}&0&0&\alpha _{-1}&1\end{bmatrix}}.}
1554:
A detailed study of the parameter dependence of the dynamics was performed by Roques and
Chekroun in. The authors observed that interaction and growth parameters leading respectively to extinction of three species, or coexistence of two, three or four species, are for the most part arranged in
2834:. Now, instead of having to integrate the system over thousands of time steps to see if any dynamics other than a fixed point attractor exist, one need only determine if the Lyapunov function exists (note: the absence of the Lyapunov function doesn't guarantee a limit cycle, torus, or chaos).
1653:. It is much easier, however, to keep the format of the equations the same and instead modify the interaction matrix. For simplicity, consider a five species example where all of the species are aligned on a circle, and each interacts only with the two neighbors on either side with strength
2907:
It is also possible to arrange the species into a line. The interaction matrix for this system is very similar to that of a circle except the interaction terms in the lower left and upper right of the matrix are deleted (those that describe the interactions between species 1 and
3155:{\displaystyle \alpha _{ij}={\begin{bmatrix}1&\alpha _{1}&0&0&0\\\alpha _{-1}&1&\alpha _{1}&0&0\\0&\alpha _{-1}&1&\alpha _{1}&0\\0&0&\alpha _{-1}&1&\alpha _{1}\\0&0&0&\alpha _{-1}&1\end{bmatrix}}}
1357:
of 0.0203. From the theorems by Hirsch, it is one of the lowest-dimensional chaotic competitive Lotka–Volterra systems. The Kaplan–Yorke dimension, a measure of the dimensionality of the attractor, is 2.074. This value is not a whole number, indicative of the
2082:
499:-values are positive. Also, note that each species can have its own growth rate and carrying capacity. A complete classification of this dynamics, even for all sign patterns of above coefficients, is available, which is based upon equivalence to the 3-type
3175:
shape. The eigenvalues from a short line form a sideways Y, but those of a long line begin to resemble the trefoil shape of the circle. This could be due to the fact that a long line is indistinguishable from a circle to those species far from the ends.
671:
1391:
467:{\displaystyle {\begin{aligned}{dx_{1} \over dt}&=r_{1}x_{1}\left(1-\left({x_{1}+\alpha _{12}x_{2} \over K_{1}}\right)\right)\\{dx_{2} \over dt}&=r_{2}x_{2}\left(1-\left({x_{2}+\alpha _{21}x_{1} \over K_{2}}\right)\right).\end{aligned}}}
1546:
eigenvalue pair. If the real part were negative, this point would be stable and the orbit would attract asymptotically. The transition between these two states, where the real part of the complex eigenvalue pair is equal to zero, is called a
798:
494:
represents the effect species 1 has on the population of species 2. These values do not have to be equal. Because this is the competitive version of the model, all interactions must be harmful (competition) and therefore all
2146:-torus are zero while that of a strange attractor is positive). If the derivative is less than zero everywhere except the equilibrium point, then the equilibrium point is a stable fixed point attractor. When searching a
1069:
1291:{\displaystyle r={\begin{bmatrix}1\\0.72\\1.53\\1.27\end{bmatrix}}\quad \alpha ={\begin{bmatrix}1&1.09&1.52&0\\0&1&0.44&1.36\\2.33&0&1&0.47\\1.21&0.51&0.35&1\end{bmatrix}}}
2239:
196:
1960:
1084:
systems Lotka–Volterra models are either unstable or have low connectivity. Kondoh and
Ackland and Gallagher have independently shown that large, stable Lotka–Volterra systems arise if the elements of
142:
54:
for predation in that the equation for each species has one term for self-interaction and one term for the interaction with other species. In the equations for predation, the base population model is
533:
2409:
676:
2301:
188:, with logistic dynamics, the Lotka–Volterra formulation adds an additional term to account for the species' interactions. Thus the competitive Lotka–Volterra equations are:
3164:
This change eliminates the
Lyapunov function described above for the system on a circle, but most likely there are other Lyapunov functions that have not been discovered.
2887:
then the real part of one of the complex eigenvalue pair becomes positive and there is a strange attractor. The disappearance of this
Lyapunov function coincides with a
1331:
970:
839:). If it is also assumed that the population of any species will increase in the absence of competition unless the population is already at the carrying capacity (
1351:
1518:{\displaystyle {\overline {x}}=\left(\alpha \right)^{-1}{\begin{bmatrix}1\\1\\1\\1\end{bmatrix}}={\begin{bmatrix}0.3013\\0.4586\\0.1307\\0.3557\end{bmatrix}}.}
929:
dimensions, and chaos cannot occur in less than three dimensions. So, Hirsch proved that competitive Lotka–Volterra systems cannot exhibit a limit cycle for
673:
or, if the carrying capacity is pulled into the interaction matrix (this doesn't actually change the equations, only how the interaction matrix is defined),
1671:
respectively. Thus, species 3 interacts only with species 2 and 4, species 1 interacts only with species 2 and 5, etc. The interaction matrix will now be
3430:
Hirsch, Morris W. (1990). "Systems of
Differential Equations That are Competitive or Cooperative. IV: Structural Stability in Three-Dimensional Systems".
2150:
for non-fixed point attractors, the existence of a
Lyapunov function can help eliminate regions of parameter space where these dynamics are impossible.
2164:
1641:. Therefore, if the competitive Lotka–Volterra equations are to be used for modeling such a system, they must incorporate this spatial structure.
35:
3570:
Vano, J A; Wildenberg, J C; Anderson, M B; Noel, J K; Sprott, J C (2006-09-15). "Chaos in low-dimensional Lotka–Volterra models of competition".
72:
1571:
An illustration of spatial structure in nature. The strength of the interaction between bee colonies is a function of their proximity. Colonies
2118:. It is often useful to imagine a Lyapunov function as the energy of the system. If the derivative of the function is equal to zero for some
1535:
1071:
and is a global attractor of every point excluding the origin. This carrying simplex contains all of the asymptotic dynamics of the system.
514:
This model can be generalized to any number of species competing against each other. One can think of the populations and growth rates as
3516:
Ackland, G. J.; Gallagher, I. D. (2004-10-08). "Stabilization of Large
Generalized Lotka-Volterra Foodwebs By Evolutionary Feedback".
1649:
One possible way to incorporate this spatial structure is to modify the nature of the Lotka–Volterra equations to something like a
1650:
821:
The definition of a competitive Lotka–Volterra system assumes that all values in the interaction matrix are positive or 0 (
51:
20:
3821:
3826:
3757:
Wildenberg, J.C.; Vano, J.A.; Sprott, J.C. (2006). "Complex spatiotemporal dynamics in Lotka–Volterra ring systems".
2077:{\displaystyle {\overline {x}}_{i}={\frac {1}{\sum _{j=1}^{N}\alpha _{ij}}}={\frac {1}{\alpha _{-1}+1+\alpha _{1}}}.}
3836:
3831:
1941:
If each species is identical in its interactions with neighboring species, then each row of the matrix is just a
884:
1945:
of the first row. A simple, but non-realistic, example of this type of system has been characterized by Sprott
3465:
Kondoh, M. (2003-02-28). "Foraging
Adaptation and the Relationship Between Food-Web Complexity and Stability".
666:{\displaystyle {\frac {dx_{i}}{dt}}=r_{i}x_{i}\left(1-{\frac {\sum _{j=1}^{N}\alpha _{ij}x_{j}}{K_{i}}}\right)}
3709:
Sprott, J.C.; Wildenberg, J.C.; Azizi, Yousef (2005). "A simple spatiotemporal chaotic Lotka–Volterra model".
3625:
949:
3795:, 1988. The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, U.K, p. 352.
2259:
2096:
3338:"Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere"
3811:
1379:
2899:
1567:
875:
Smale showed that Lotka–Volterra systems that meet the above conditions and have five or more species (
3718:
3675:
3579:
3525:
3388:
1371:
1122:
A simple 4-dimensional example of a competitive Lotka–Volterra system has been characterized by Vano
523:
55:
1610:
500:
31:
1542:, −0.3342, and −1.0319. This point is unstable due to the positive value of the real part of the
3816:
3603:
3498:
3412:
3318:
3267:
3221:
2153:
The spatial system introduced above has a
Lyapunov function that has been explored by Wildenberg
2115:
515:
3377:"Systems of differential equations which are competitive or cooperative: III. Competing species"
3774:
3734:
3691:
3648:
3595:
3549:
3541:
3490:
3482:
3447:
3404:
3357:
3310:
3302:
3259:
3213:
2139:
2135:
2123:
2092:
1950:
1556:
1375:
1367:
1363:
1354:
899:
793:{\displaystyle {\frac {dx_{i}}{dt}}=r_{i}x_{i}\left(1-\sum _{j=1}^{N}\alpha _{ij}x_{j}\right)}
159:
59:
3196:(1983). "Lotka-Volterra equation and replicator dynamics: A two-dimensional classification".
1303:
3766:
3726:
3683:
3640:
3587:
3533:
3474:
3439:
3396:
3349:
3294:
3251:
3205:
2888:
2158:
2147:
2119:
1548:
2157:
If all species are identical in their spatial interactions, then the interaction matrix is
921:−1. This is important because a limit cycle cannot exist in fewer than two dimensions, an
1954:
3242:(1995). "Lotka-Volterra equation and replicator dynamics: new issues in classification".
3722:
3679:
3583:
3529:
3392:
1531:
equilibrium points, but all others have at least one species' population equal to zero.
1101:
3239:
3193:
1609:
away. This doesn't mean, however, that those far colonies can be ignored. There is a
1543:
1336:
804:
is the total number of interacting species. For simplicity all self-interacting terms
34:
of species competing for some common resource. They can be further generalised to the
3591:
19:
This article is about the competition equations. For the predator-prey equations, see
3805:
3687:
3502:
3416:
3225:
3168:
2308:
1383:
957:
3400:
3322:
3271:
3792:
3607:
2903:
The eigenvalues of a circle, short line, and long line plotted in the complex plane
1092:(i.e. the features of the species) can evolve in accordance with natural selection.
3537:
3770:
3644:
3730:
1942:
1106:
888:
3473:(5611). American Association for the Advancement of Science (AAAS): 1388–1391.
2877:
then all eigenvalues are negative and the only attractor is a fixed point. If
2327:
The Lyapunov function exists if the real part of the eigenvalues are positive (
1064:{\displaystyle \Delta _{N-1}=\left\{x_{i}:x_{i}\geq 0,\sum _{i}x_{i}=1\right\}}
941:< 4. This is still in agreement with Smale that any dynamics can occur for
853:), then some definite statements can be made about the behavior of the system.
3778:
3738:
3695:
3652:
3599:
3545:
3486:
3451:
3408:
3361:
3306:
3285:
Smale, S. (1976). "On the differential equations of species in competition".
3263:
3217:
857:
The populations of all species will be bounded between 0 and 1 at all times (
3478:
3376:
3337:
2127:
914:
880:
3553:
3494:
1370:, the point at which all derivatives are equal to zero but that is not the
3314:
3438:(5). Society for Industrial & Applied Mathematics (SIAM): 1225–1234.
906:
39:
3666:
Nese, Jon M. (1989). "Quantifying local predictability in phase space".
3348:(3). Society for Industrial & Applied Mathematics (SIAM): 423–439.
3298:
3255:
3209:
3172:
1359:
965:
485:
represents the effect species 2 has on the population of species 1 and
66:
3443:
3353:
905:
Hirsch proved that all of the dynamics of the attractor occur on a
2898:
1566:
1100:
934:
892:
2234:{\displaystyle \lambda _{k}=\sum _{j=0}^{N-1}c_{j}\gamma ^{kj}}
1595:
do not interact directly, but affect each other through colony
3626:"Probing chaos and biodiversity in a simple competition model"
1126:
Here the growth rates and interaction matrix have been set to
137:{\displaystyle {dx \over dt}=rx\left(1-{x \over K}\right).}
913:−1. This essentially says that the attractor cannot have
3250:(5). Springer Science and Business Media LLC: 447–453.
3204:(3). Springer Science and Business Media LLC: 201–211.
2942:
2161:. The eigenvalues of a circulant matrix are given by
1953:
for these systems has a very simple form given by the
1699:
1477:
1434:
1193:
1146:
1080:
matrix must have all positive eigenvalues. For large-
2920:
2407:
2262:
2167:
1963:
1677:
1613:
effect that permeates through the system. If colony
1394:
1339:
1306:
1134:
973:
679:
536:
194:
75:
3167:
The eigenvalues of the circle system plotted in the
3293:(1). Springer Science and Business Media LLC: 5–7.
2324:th value in the first row of the circulant matrix.
3154:
2815:
2295:
2233:
2076:
1931:
1517:
1345:
1325:
1290:
1063:
872:) as long as the populations started out positive.
792:
665:
466:
136:
1105:The competitive Lotka–Volterra system plotted in
3619:
3617:
3524:(15). American Physical Society (APS): 158701.
150:is the size of the population at a given time,
3752:
3750:
3748:
3624:Roques, Lionel; Chekroun, Mickaël D. (2011).
1538:of the system at this point are 0.0414±0.1903
948:More specifically, Hirsch showed there is an
8:
3565:
3563:
65:The logistic population model, when used by
1353:. This system is chaotic and has a largest
16:Model of multi-species population dynamics
3130:
3101:
3081:
3052:
3032:
3003:
2983:
2954:
2937:
2925:
2919:
2781:
2762:
2746:
2714:
2695:
2679:
2641:
2628:
2618:
2601:
2588:
2578:
2561:
2533:
2520:
2503:
2475:
2462:
2425:
2408:
2406:
2283:
2273:
2261:
2222:
2212:
2196:
2185:
2172:
2166:
2130:, but it must be either a limit cycle or
2114:whose existence in a system demonstrates
2062:
2040:
2030:
2015:
2005:
1994:
1984:
1975:
1965:
1962:
1904:
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1848:
1819:
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1733:
1711:
1694:
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1420:
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1338:
1311:
1305:
1188:
1141:
1133:
1044:
1034:
1015:
1002:
978:
972:
779:
766:
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581:
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247:
237:
209:
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195:
193:
116:
76:
74:
154:is inherent per-capita growth rate, and
3185:
58:. For the competition equations, the
3432:SIAM Journal on Mathematical Analysis
3342:SIAM Journal on Mathematical Analysis
526:. Then the equation for any species
7:
2296:{\displaystyle \gamma =e^{i2\pi /N}}
28:competitive Lotka–Volterra equations
2401:. The Lyapunov function exists if
36:generalized Lotka–Volterra equation
1076:To create a stable ecosystem the α
975:
14:
3578:(10). IOP Publishing: 2391–2404.
925:-torus cannot exist in less than
69:often takes the following form:
3287:Journal of Mathematical Biology
1181:
3711:Chaos, Solitons & Fractals
3668:Physica D: Nonlinear Phenomena
2768:
2739:
2701:
2672:
2558:
2546:
2500:
2488:
2431:
2418:
2349:). Consider the system where
2126:, then that orbit is a stable
1118:value represented by the color
1:
3717:(4). Elsevier BV: 1035–1043.
3674:(1–2). Elsevier BV: 237–250.
3538:10.1103/physrevlett.93.158701
2138:(this is because the largest
3771:10.1016/j.ecocom.2005.12.001
3688:10.1016/0167-2789(89)90105-x
3645:10.1016/j.ecocom.2010.08.004
3387:(1). IOP Publishing: 51–71.
2895:Line systems and eigenvalues
1970:
1400:
3765:(2). Elsevier BV: 140–147.
3731:10.1016/j.chaos.2005.02.015
3592:10.1088/0951-7715/19/10/006
1527:Note that there are always
1378:the interaction matrix and
50:The form is similar to the
3853:
3639:(1). Elsevier BV: 98–104.
3375:Hirsch, M W (1988-02-01).
3336:Hirsch, Morris W. (1985).
30:are a simple model of the
18:
3401:10.1088/0951-7715/1/1/003
1651:reaction–diffusion system
1579:interact, as do colonies
1559:induced by local chaos.
1362:structure inherent in a
52:Lotka–Volterra equations
21:Lotka–Volterra equations
3518:Physical Review Letters
3479:10.1126/science.1079154
1326:{\displaystyle K_{i}=1}
170:Given two populations,
3244:Biological Cybernetics
3198:Biological Cybernetics
3156:
2904:
2817:
2297:
2235:
2207:
2078:
2010:
1957:of the sum of the row
1933:
1617:interacts with colony
1600:
1519:
1347:
1327:
1292:
1119:
1065:
883:behavior, including a
794:
761:
667:
621:
468:
138:
3759:Ecological Complexity
3633:Ecological Complexity
3157:
2902:
2818:
2298:
2236:
2181:
2142:of a limit cycle and
2079:
1990:
1934:
1570:
1520:
1348:
1328:
1293:
1104:
1097:4-dimensional example
1066:
879:≥ 5) can exhibit any
795:
741:
668:
601:
469:
139:
2918:
2405:
2260:
2165:
1961:
1675:
1563:Spatial arrangements
1392:
1337:
1304:
1132:
971:
813:are often set to 1.
677:
534:
192:
73:
40:trophic interactions
3822:Population dynamics
3723:2005CSF....26.1035S
3680:1989PhyD...35..237N
3584:2006Nonli..19.2391V
3530:2004PhRvL..93o8701A
3393:1988Nonli...1...51H
2134:-torus - but not a
1645:Matrix organization
501:replicator equation
32:population dynamics
3827:Population ecology
3299:10.1007/bf00307854
3256:10.1007/bf00201420
3240:Bomze, Immanuel M.
3210:10.1007/bf00318088
3194:Bomze, Immanuel M.
3152:
3146:
2905:
2813:
2811:
2293:
2231:
2122:not including the
2087:Lyapunov functions
2074:
1929:
1920:
1601:
1557:species abundances
1515:
1506:
1463:
1386:, and is equal to
1374:, can be found by
1366:. The coexisting
1343:
1323:
1288:
1282:
1175:
1120:
1061:
1039:
790:
663:
464:
462:
134:
3837:Population models
3832:Community ecology
2797:
2730:
2140:Lyapunov exponent
2136:strange attractor
2124:equilibrium point
2093:Lyapunov function
2069:
2025:
1973:
1951:equilibrium point
1403:
1368:equilibrium point
1364:strange attractor
1355:Lyapunov exponent
1346:{\displaystyle i}
1030:
817:Possible dynamics
705:
656:
562:
446:
355:
315:
224:
160:carrying capacity
124:
94:
60:logistic equation
3844:
3796:
3789:
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3611:
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3558:
3557:
3513:
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3506:
3462:
3456:
3455:
3427:
3421:
3420:
3372:
3366:
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3333:
3327:
3326:
3282:
3276:
3275:
3236:
3230:
3229:
3190:
3161:
3159:
3158:
3153:
3151:
3150:
3138:
3137:
3106:
3105:
3089:
3088:
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3056:
3040:
3039:
3008:
3007:
2991:
2990:
2959:
2958:
2933:
2932:
2889:Hopf bifurcation
2886:
2876:
2866:
2856:
2846:
2833:
2822:
2820:
2819:
2814:
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2802:
2798:
2793:
2782:
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2766:
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2753:
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2726:
2715:
2700:
2699:
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2686:
2659:
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2649:
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2622:
2610:
2609:
2605:
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2582:
2570:
2569:
2565:
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2527:
2512:
2511:
2507:
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2430:
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2400:
2387:
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2361:
2348:
2337:
2319:
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2300:
2299:
2294:
2292:
2291:
2287:
2255:
2240:
2238:
2237:
2232:
2230:
2229:
2217:
2216:
2206:
2195:
2177:
2176:
2148:dynamical system
2113:
2083:
2081:
2080:
2075:
2070:
2068:
2067:
2066:
2048:
2047:
2031:
2026:
2024:
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2022:
2009:
2004:
1985:
1980:
1979:
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1966:
1938:
1936:
1935:
1930:
1925:
1924:
1912:
1911:
1887:
1886:
1873:
1872:
1856:
1855:
1824:
1823:
1807:
1806:
1775:
1774:
1758:
1757:
1741:
1740:
1716:
1715:
1690:
1689:
1670:
1661:
1549:Hopf bifurcation
1530:
1524:
1522:
1521:
1516:
1511:
1510:
1468:
1467:
1428:
1427:
1419:
1404:
1396:
1352:
1350:
1349:
1344:
1332:
1330:
1329:
1324:
1316:
1315:
1297:
1295:
1294:
1289:
1287:
1286:
1180:
1179:
1117:
1091:
1070:
1068:
1067:
1062:
1060:
1056:
1049:
1048:
1038:
1020:
1019:
1007:
1006:
989:
988:
964:−1)-dimensional
871:
867:
852:
848:
838:
834:
830:
812:
803:
799:
797:
796:
791:
789:
785:
784:
783:
774:
773:
760:
755:
729:
728:
719:
718:
706:
704:
696:
695:
694:
681:
672:
670:
669:
664:
662:
658:
657:
655:
654:
645:
644:
643:
634:
633:
620:
615:
599:
586:
585:
576:
575:
563:
561:
553:
552:
551:
538:
529:
521:
493:
484:
473:
471:
470:
465:
463:
456:
452:
451:
447:
445:
444:
435:
434:
433:
424:
423:
411:
410:
400:
383:
382:
373:
372:
356:
354:
346:
345:
344:
331:
325:
321:
320:
316:
314:
313:
304:
303:
302:
293:
292:
280:
279:
269:
252:
251:
242:
241:
225:
223:
215:
214:
213:
200:
187:
178:
157:
153:
149:
143:
141:
140:
135:
130:
126:
125:
117:
95:
93:
85:
77:
3852:
3851:
3847:
3846:
3845:
3843:
3842:
3841:
3802:
3801:
3800:
3799:
3790:
3786:
3756:
3755:
3746:
3708:
3707:
3703:
3665:
3664:
3660:
3628:
3623:
3622:
3615:
3569:
3568:
3561:
3515:
3514:
3510:
3464:
3463:
3459:
3444:10.1137/0521067
3429:
3428:
3424:
3374:
3373:
3369:
3354:10.1137/0516030
3335:
3334:
3330:
3284:
3283:
3279:
3238:
3237:
3233:
3192:
3191:
3187:
3182:
3145:
3144:
3139:
3126:
3124:
3119:
3114:
3108:
3107:
3097:
3095:
3090:
3077:
3075:
3070:
3064:
3063:
3058:
3048:
3046:
3041:
3028:
3026:
3020:
3019:
3014:
3009:
2999:
2997:
2992:
2979:
2976:
2975:
2970:
2965:
2960:
2950:
2948:
2938:
2921:
2916:
2915:
2897:
2884:
2878:
2874:
2868:
2864:
2858:
2854:
2848:
2844:
2838:
2824:
2810:
2809:
2783:
2777:
2758:
2742:
2716:
2710:
2691:
2675:
2657:
2656:
2624:
2614:
2584:
2574:
2529:
2516:
2471:
2458:
2451:
2447:
2434:
2421:
2403:
2402:
2395:
2389:
2382:
2376:
2369:
2363:
2356:
2350:
2339:
2334:
2328:
2317:
2312:
2269:
2258:
2257:
2254:
2244:
2218:
2208:
2168:
2163:
2162:
2100:
2089:
2058:
2036:
2035:
2011:
1989:
1964:
1959:
1958:
1949:The coexisting
1919:
1918:
1913:
1900:
1898:
1893:
1888:
1878:
1875:
1874:
1864:
1862:
1857:
1844:
1842:
1837:
1831:
1830:
1825:
1815:
1813:
1808:
1795:
1793:
1787:
1786:
1781:
1776:
1766:
1764:
1759:
1746:
1743:
1742:
1729:
1727:
1722:
1717:
1707:
1705:
1695:
1678:
1673:
1672:
1669:
1663:
1660:
1654:
1647:
1606:
1565:
1528:
1505:
1504:
1498:
1497:
1491:
1490:
1484:
1483:
1473:
1462:
1461:
1455:
1454:
1448:
1447:
1441:
1440:
1430:
1409:
1408:
1390:
1389:
1335:
1334:
1307:
1302:
1301:
1281:
1280:
1275:
1270:
1265:
1259:
1258:
1253:
1248:
1243:
1237:
1236:
1231:
1226:
1221:
1215:
1214:
1209:
1204:
1199:
1189:
1174:
1173:
1167:
1166:
1160:
1159:
1153:
1152:
1142:
1130:
1129:
1116:
1110:
1099:
1090:
1086:
1079:
1040:
1011:
998:
997:
993:
974:
969:
968:
933:< 3, or any
869:
864:
858:
850:
845:
840:
836:
832:
827:
822:
819:
811:
805:
801:
775:
762:
734:
730:
720:
710:
697:
686:
682:
675:
674:
646:
635:
622:
600:
591:
587:
577:
567:
554:
543:
539:
532:
531:
527:
519:
512:
492:
486:
483:
477:
461:
460:
436:
425:
415:
402:
401:
395:
388:
384:
374:
364:
357:
347:
336:
332:
327:
326:
305:
294:
284:
271:
270:
264:
257:
253:
243:
233:
226:
216:
205:
201:
190:
189:
186:
180:
177:
171:
168:
155:
151:
147:
109:
105:
86:
78:
71:
70:
48:
24:
17:
12:
11:
5:
3850:
3848:
3840:
3839:
3834:
3829:
3824:
3819:
3814:
3804:
3803:
3798:
3797:
3791:Hofbauer, J.,
3784:
3744:
3701:
3658:
3613:
3559:
3508:
3457:
3422:
3367:
3328:
3277:
3231:
3184:
3183:
3181:
3178:
3149:
3143:
3140:
3136:
3133:
3129:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3109:
3104:
3100:
3096:
3094:
3091:
3087:
3084:
3080:
3076:
3074:
3071:
3069:
3066:
3065:
3062:
3059:
3055:
3051:
3047:
3045:
3042:
3038:
3035:
3031:
3027:
3025:
3022:
3021:
3018:
3015:
3013:
3010:
3006:
3002:
2998:
2996:
2993:
2989:
2986:
2982:
2978:
2977:
2974:
2971:
2969:
2966:
2964:
2961:
2957:
2953:
2949:
2947:
2944:
2943:
2941:
2936:
2931:
2928:
2924:
2896:
2893:
2882:
2872:
2862:
2852:
2842:
2808:
2805:
2801:
2796:
2792:
2789:
2786:
2780:
2776:
2773:
2770:
2765:
2761:
2757:
2752:
2749:
2745:
2741:
2738:
2734:
2729:
2725:
2722:
2719:
2713:
2709:
2706:
2703:
2698:
2694:
2690:
2685:
2682:
2678:
2674:
2671:
2668:
2665:
2662:
2660:
2658:
2654:
2648:
2644:
2640:
2637:
2634:
2631:
2627:
2621:
2617:
2613:
2608:
2604:
2600:
2597:
2594:
2591:
2587:
2581:
2577:
2573:
2568:
2564:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2532:
2526:
2523:
2519:
2515:
2510:
2506:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2474:
2468:
2465:
2461:
2457:
2454:
2450:
2446:
2443:
2440:
2437:
2435:
2433:
2428:
2424:
2420:
2417:
2414:
2411:
2410:
2393:
2380:
2367:
2354:
2332:
2315:
2290:
2286:
2282:
2279:
2276:
2272:
2268:
2265:
2249:
2228:
2225:
2221:
2215:
2211:
2205:
2202:
2199:
2194:
2191:
2188:
2184:
2180:
2175:
2171:
2099:of the system
2088:
2085:
2073:
2065:
2061:
2057:
2054:
2051:
2046:
2043:
2039:
2034:
2029:
2021:
2018:
2014:
2008:
2003:
2000:
1997:
1993:
1988:
1983:
1978:
1972:
1969:
1928:
1923:
1917:
1914:
1910:
1907:
1903:
1899:
1897:
1894:
1892:
1889:
1885:
1881:
1877:
1876:
1871:
1867:
1863:
1861:
1858:
1854:
1851:
1847:
1843:
1841:
1838:
1836:
1833:
1832:
1829:
1826:
1822:
1818:
1814:
1812:
1809:
1805:
1802:
1798:
1794:
1792:
1789:
1788:
1785:
1782:
1780:
1777:
1773:
1769:
1765:
1763:
1760:
1756:
1753:
1749:
1745:
1744:
1739:
1736:
1732:
1728:
1726:
1723:
1721:
1718:
1714:
1710:
1706:
1704:
1701:
1700:
1698:
1693:
1688:
1685:
1681:
1667:
1658:
1646:
1643:
1605:
1602:
1564:
1561:
1514:
1509:
1503:
1500:
1499:
1496:
1493:
1492:
1489:
1486:
1485:
1482:
1479:
1478:
1476:
1471:
1466:
1460:
1457:
1456:
1453:
1450:
1449:
1446:
1443:
1442:
1439:
1436:
1435:
1433:
1426:
1423:
1418:
1415:
1412:
1407:
1402:
1399:
1342:
1322:
1319:
1314:
1310:
1285:
1279:
1276:
1274:
1271:
1269:
1266:
1264:
1261:
1260:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1238:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1216:
1213:
1210:
1208:
1205:
1203:
1200:
1198:
1195:
1194:
1192:
1187:
1184:
1178:
1172:
1169:
1168:
1165:
1162:
1161:
1158:
1155:
1154:
1151:
1148:
1147:
1145:
1140:
1137:
1114:
1098:
1095:
1094:
1093:
1088:
1077:
1074:
1073:
1072:
1059:
1055:
1052:
1047:
1043:
1037:
1033:
1029:
1026:
1023:
1018:
1014:
1010:
1005:
1001:
996:
992:
987:
984:
981:
977:
903:
873:
862:
843:
825:
818:
815:
809:
788:
782:
778:
772:
769:
765:
759:
754:
751:
748:
744:
740:
737:
733:
727:
723:
717:
713:
709:
703:
700:
693:
689:
685:
661:
653:
649:
642:
638:
632:
629:
625:
619:
614:
611:
608:
604:
597:
594:
590:
584:
580:
574:
570:
566:
560:
557:
550:
546:
542:
511:
505:
490:
481:
459:
455:
450:
443:
439:
432:
428:
422:
418:
414:
409:
405:
398:
394:
391:
387:
381:
377:
371:
367:
363:
360:
358:
353:
350:
343:
339:
335:
329:
328:
324:
319:
312:
308:
301:
297:
291:
287:
283:
278:
274:
267:
263:
260:
256:
250:
246:
240:
236:
232:
229:
227:
222:
219:
212:
208:
204:
198:
197:
184:
175:
167:
164:
133:
129:
123:
120:
115:
112:
108:
104:
101:
98:
92:
89:
84:
81:
62:is the basis.
47:
44:
15:
13:
10:
9:
6:
4:
3:
2:
3849:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3820:
3818:
3815:
3813:
3810:
3809:
3807:
3794:
3788:
3785:
3780:
3776:
3772:
3768:
3764:
3760:
3753:
3751:
3749:
3745:
3740:
3736:
3732:
3728:
3724:
3720:
3716:
3712:
3705:
3702:
3697:
3693:
3689:
3685:
3681:
3677:
3673:
3669:
3662:
3659:
3654:
3650:
3646:
3642:
3638:
3634:
3627:
3620:
3618:
3614:
3609:
3605:
3601:
3597:
3593:
3589:
3585:
3581:
3577:
3573:
3566:
3564:
3560:
3555:
3551:
3547:
3543:
3539:
3535:
3531:
3527:
3523:
3519:
3512:
3509:
3504:
3500:
3496:
3492:
3488:
3484:
3480:
3476:
3472:
3468:
3461:
3458:
3453:
3449:
3445:
3441:
3437:
3433:
3426:
3423:
3418:
3414:
3410:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3378:
3371:
3368:
3363:
3359:
3355:
3351:
3347:
3343:
3339:
3332:
3329:
3324:
3320:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3281:
3278:
3273:
3269:
3265:
3261:
3257:
3253:
3249:
3245:
3241:
3235:
3232:
3227:
3223:
3219:
3215:
3211:
3207:
3203:
3199:
3195:
3189:
3186:
3179:
3177:
3174:
3170:
3169:complex plane
3165:
3162:
3147:
3141:
3134:
3131:
3127:
3121:
3116:
3111:
3102:
3098:
3092:
3085:
3082:
3078:
3072:
3067:
3060:
3053:
3049:
3043:
3036:
3033:
3029:
3023:
3016:
3011:
3004:
3000:
2994:
2987:
2984:
2980:
2972:
2967:
2962:
2955:
2951:
2945:
2939:
2934:
2929:
2926:
2922:
2913:
2911:
2901:
2894:
2892:
2890:
2881:
2871:
2861:
2851:
2841:
2837:Example: Let
2835:
2831:
2827:
2806:
2803:
2799:
2794:
2790:
2787:
2784:
2778:
2774:
2771:
2763:
2759:
2755:
2750:
2747:
2743:
2736:
2732:
2727:
2723:
2720:
2717:
2711:
2707:
2704:
2696:
2692:
2688:
2683:
2680:
2676:
2669:
2666:
2663:
2661:
2652:
2646:
2642:
2638:
2635:
2632:
2629:
2625:
2619:
2615:
2611:
2606:
2602:
2598:
2595:
2592:
2589:
2585:
2579:
2575:
2571:
2566:
2562:
2555:
2552:
2549:
2543:
2540:
2537:
2534:
2530:
2524:
2521:
2517:
2513:
2508:
2504:
2497:
2494:
2491:
2485:
2482:
2479:
2476:
2472:
2466:
2463:
2459:
2455:
2452:
2448:
2444:
2441:
2438:
2436:
2426:
2422:
2415:
2412:
2399:
2392:
2386:
2379:
2373:
2366:
2360:
2353:
2346:
2342:
2335:
2325:
2323:
2318:
2310:
2309:root of unity
2306:
2288:
2284:
2280:
2277:
2274:
2270:
2266:
2263:
2252:
2247:
2241:
2226:
2223:
2219:
2213:
2209:
2203:
2200:
2197:
2192:
2189:
2186:
2182:
2178:
2173:
2169:
2160:
2156:
2151:
2149:
2145:
2141:
2137:
2133:
2129:
2125:
2121:
2117:
2111:
2107:
2103:
2098:
2094:
2086:
2084:
2071:
2063:
2059:
2055:
2052:
2049:
2044:
2041:
2037:
2032:
2027:
2019:
2016:
2012:
2006:
2001:
1998:
1995:
1991:
1986:
1981:
1976:
1967:
1956:
1952:
1948:
1944:
1939:
1926:
1921:
1915:
1908:
1905:
1901:
1895:
1890:
1883:
1879:
1869:
1865:
1859:
1852:
1849:
1845:
1839:
1834:
1827:
1820:
1816:
1810:
1803:
1800:
1796:
1790:
1783:
1778:
1771:
1767:
1761:
1754:
1751:
1747:
1737:
1734:
1730:
1724:
1719:
1712:
1708:
1702:
1696:
1691:
1686:
1683:
1679:
1666:
1657:
1652:
1644:
1642:
1640:
1636:
1632:
1628:
1624:
1620:
1616:
1612:
1603:
1598:
1594:
1590:
1586:
1582:
1578:
1574:
1569:
1562:
1560:
1558:
1552:
1550:
1545:
1541:
1537:
1532:
1525:
1512:
1507:
1501:
1494:
1487:
1480:
1474:
1469:
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1458:
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1437:
1431:
1424:
1421:
1416:
1413:
1410:
1405:
1397:
1387:
1385:
1384:column vector
1381:
1377:
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1369:
1365:
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1356:
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1317:
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1196:
1190:
1185:
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1176:
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1156:
1149:
1143:
1138:
1135:
1127:
1125:
1113:
1108:
1103:
1096:
1083:
1075:
1057:
1053:
1050:
1045:
1041:
1035:
1031:
1027:
1024:
1021:
1016:
1012:
1008:
1003:
999:
994:
990:
985:
982:
979:
967:
963:
959:
955:
951:
947:
946:
944:
940:
937:or chaos for
936:
932:
928:
924:
920:
917:greater than
916:
912:
909:of dimension
908:
904:
901:
897:
895:
890:
886:
882:
878:
874:
865:
856:
855:
854:
846:
828:
816:
814:
808:
786:
780:
776:
770:
767:
763:
757:
752:
749:
746:
742:
738:
735:
731:
725:
721:
715:
711:
707:
701:
698:
691:
687:
683:
659:
651:
647:
640:
636:
630:
627:
623:
617:
612:
609:
606:
602:
595:
592:
588:
582:
578:
572:
568:
564:
558:
555:
548:
544:
540:
525:
517:
509:
506:
504:
502:
498:
489:
480:
474:
457:
453:
448:
441:
437:
430:
426:
420:
416:
412:
407:
403:
396:
392:
389:
385:
379:
375:
369:
365:
361:
359:
351:
348:
341:
337:
333:
322:
317:
310:
306:
299:
295:
289:
285:
281:
276:
272:
265:
261:
258:
254:
248:
244:
238:
234:
230:
228:
220:
217:
210:
206:
202:
183:
174:
165:
163:
161:
144:
131:
127:
121:
118:
113:
110:
106:
102:
99:
96:
90:
87:
82:
79:
68:
63:
61:
57:
53:
45:
43:
41:
37:
33:
29:
22:
3812:Chaotic maps
3787:
3762:
3758:
3714:
3710:
3704:
3671:
3667:
3661:
3636:
3632:
3575:
3572:Nonlinearity
3571:
3521:
3517:
3511:
3470:
3466:
3460:
3435:
3431:
3425:
3384:
3381:Nonlinearity
3380:
3370:
3345:
3341:
3331:
3290:
3286:
3280:
3247:
3243:
3234:
3201:
3197:
3188:
3166:
3163:
2914:
2909:
2906:
2879:
2869:
2859:
2849:
2839:
2836:
2829:
2825:
2397:
2390:
2384:
2377:
2371:
2364:
2358:
2351:
2344:
2340:
2330:
2326:
2321:
2313:
2304:
2250:
2245:
2242:
2154:
2152:
2143:
2131:
2109:
2105:
2101:
2090:
1946:
1940:
1664:
1655:
1648:
1638:
1634:
1630:
1626:
1622:
1618:
1614:
1607:
1596:
1592:
1588:
1584:
1580:
1576:
1572:
1553:
1539:
1533:
1526:
1388:
1382:by the unit
1299:
1128:
1123:
1121:
1111:
1081:
961:
958:homeomorphic
953:
942:
938:
930:
926:
922:
918:
910:
893:
876:
860:
841:
823:
820:
806:
513:
507:
496:
487:
478:
475:
181:
172:
169:
145:
64:
49:
27:
25:
3793:Sigmund, K.
1943:permutation
1536:eigenvalues
1380:multiplying
1107:phase space
889:limit cycle
885:fixed point
166:Two species
56:exponential
38:to include
3806:Categories
2256:and where
1611:transitive
1604:Background
900:attractors
881:asymptotic
868:, for all
67:ecologists
3817:Equations
3779:1476-945X
3739:0960-0779
3696:0167-2789
3653:1476-945X
3600:0951-7715
3546:0031-9007
3503:129162096
3487:0036-8075
3452:0036-1410
3417:250848783
3409:0951-7715
3362:0036-1410
3307:0303-6812
3264:0340-1200
3226:206774680
3218:0340-1200
3132:−
3128:α
3099:α
3083:−
3079:α
3050:α
3034:−
3030:α
3001:α
2985:−
2981:α
2952:α
2923:α
2912:, etc.).
2788:π
2775:
2760:α
2748:−
2744:α
2721:π
2708:
2693:α
2681:−
2677:α
2636:π
2616:α
2596:π
2576:α
2553:−
2541:π
2522:−
2518:α
2495:−
2483:π
2464:−
2460:α
2445:
2423:λ
2416:
2281:π
2264:γ
2220:γ
2201:−
2183:∑
2170:λ
2159:circulant
2128:attractor
2116:stability
2060:α
2042:−
2038:α
2013:α
1992:∑
1971:¯
1906:−
1902:α
1880:α
1866:α
1850:−
1846:α
1817:α
1801:−
1797:α
1768:α
1752:−
1748:α
1735:−
1731:α
1709:α
1680:α
1422:−
1414:α
1401:¯
1376:inverting
1183:α
1109:with the
1032:∑
1022:≥
983:−
976:Δ
950:invariant
915:dimension
764:α
743:∑
739:−
624:α
603:∑
596:−
417:α
393:−
286:α
262:−
114:−
3554:15524949
3495:12610303
3323:33201460
3272:18754189
2828:= 0, …,
2343:= 0, …,
2336:) > 0
2311:. Here
2097:function
1637:through
1633:affects
1333:for all
960:to the (
956:that is
907:manifold
849:for all
831:for all
530:becomes
522:'s as a
46:Overview
3719:Bibcode
3676:Bibcode
3608:9417299
3580:Bibcode
3526:Bibcode
3467:Science
3389:Bibcode
3315:1022822
3173:trefoil
3171:form a
2885:= 0.852
2865:= 0.237
2845:= 0.451
2320:is the
1955:inverse
1629:, then
1544:complex
1360:fractal
966:simplex
516:vectors
510:species
158:is the
3777:
3737:
3694:
3651:
3606:
3598:
3552:
3544:
3501:
3493:
3485:
3450:
3415:
3407:
3360:
3321:
3313:
3305:
3270:
3262:
3224:
3216:
2867:. If
2857:, and
2388:, and
2155:et al.
1947:et al.
1621:, and
1502:0.3557
1495:0.1307
1488:0.4586
1481:0.3013
1372:origin
1124:et al.
896:-torus
847:> 0
800:where
524:matrix
476:Here,
3629:(PDF)
3604:S2CID
3499:S2CID
3413:S2CID
3319:S2CID
3268:S2CID
3222:S2CID
3180:Notes
2875:= 0.5
2855:= 0.5
2120:orbit
2095:is a
1625:with
1300:with
945:≥ 5.
935:torus
898:, or
891:, an
146:Here
3775:ISSN
3735:ISSN
3692:ISSN
3649:ISSN
3596:ISSN
3550:PMID
3542:ISSN
3491:PMID
3483:ISSN
3448:ISSN
3405:ISSN
3358:ISSN
3311:PMID
3303:ISSN
3260:ISSN
3214:ISSN
2823:for
2804:>
2338:for
2303:the
2243:for
1662:and
1591:and
1583:and
1575:and
1534:The
1273:0.35
1268:0.51
1263:1.21
1256:0.47
1241:2.33
1234:1.36
1229:0.44
1207:1.52
1202:1.09
1171:1.27
1164:1.53
1157:0.72
952:set
887:, a
859:0 ≤
179:and
26:The
3767:doi
3727:doi
3684:doi
3641:doi
3588:doi
3534:doi
3475:doi
3471:299
3440:doi
3397:doi
3350:doi
3295:doi
3252:doi
3206:doi
2832:− 1
2772:cos
2705:cos
2329:Re(
2307:th
2253:− 1
2248:= 0
866:≤ 1
829:≥ 0
3808::
3773:.
3761:.
3747:^
3733:.
3725:.
3715:26
3713:.
3690:.
3682:.
3672:35
3670:.
3647:.
3635:.
3631:.
3616:^
3602:.
3594:.
3586:.
3576:19
3574:.
3562:^
3548:.
3540:.
3532:.
3522:93
3520:.
3497:.
3489:.
3481:.
3469:.
3446:.
3436:21
3434:.
3411:.
3403:.
3395:.
3383:.
3379:.
3356:.
3346:16
3344:.
3340:.
3317:.
3309:.
3301:.
3289:.
3266:.
3258:.
3248:72
3246:.
3220:.
3212:.
3202:48
3200:.
2891:.
2853:−1
2847:,
2843:−2
2442:Re
2413:Re
2396:=
2383:=
2375:,
2370:=
2368:−1
2362:,
2357:=
2355:−2
2347:/2
2104:=
2091:A
1659:−1
1587:.
1551:.
1089:ij
1078:ij
835:,
826:ij
810:ii
518:,
503:.
491:21
482:12
421:21
290:12
162:.
42:.
3781:.
3769::
3763:3
3741:.
3729::
3721::
3698:.
3686::
3678::
3655:.
3643::
3637:8
3610:.
3590::
3582::
3556:.
3536::
3528::
3505:.
3477::
3454:.
3442::
3419:.
3399::
3391::
3385:1
3364:.
3352::
3325:.
3297::
3291:3
3274:.
3254::
3228:.
3208::
3148:]
3142:1
3135:1
3122:0
3117:0
3112:0
3103:1
3093:1
3086:1
3073:0
3068:0
3061:0
3054:1
3044:1
3037:1
3024:0
3017:0
3012:0
3005:1
2995:1
2988:1
2973:0
2968:0
2963:0
2956:1
2946:1
2940:[
2935:=
2930:j
2927:i
2910:N
2883:1
2880:α
2873:1
2870:α
2863:2
2860:α
2850:α
2840:α
2830:N
2826:k
2807:0
2800:)
2795:N
2791:k
2785:2
2779:(
2769:)
2764:1
2756:+
2751:1
2740:(
2737:+
2733:)
2728:N
2724:k
2718:4
2712:(
2702:)
2697:2
2689:+
2684:2
2673:(
2670:+
2667:1
2664:=
2653:)
2647:N
2643:/
2639:k
2633:4
2630:i
2626:e
2620:2
2612:+
2607:N
2603:/
2599:k
2593:2
2590:i
2586:e
2580:1
2572:+
2567:N
2563:/
2559:)
2556:1
2550:N
2547:(
2544:k
2538:2
2535:i
2531:e
2525:1
2514:+
2509:N
2505:/
2501:)
2498:2
2492:N
2489:(
2486:k
2480:2
2477:i
2473:e
2467:2
2456:+
2453:1
2449:(
2439:=
2432:)
2427:k
2419:(
2398:d
2394:2
2391:α
2385:c
2381:1
2378:α
2372:b
2365:α
2359:a
2352:α
2345:N
2341:k
2333:k
2331:λ
2322:j
2316:j
2314:c
2305:N
2289:N
2285:/
2278:2
2275:i
2271:e
2267:=
2251:N
2246:k
2227:j
2224:k
2214:j
2210:c
2204:1
2198:N
2193:0
2190:=
2187:j
2179:=
2174:k
2144:n
2132:n
2112:)
2110:x
2108:(
2106:f
2102:f
2072:.
2064:1
2056:+
2053:1
2050:+
2045:1
2033:1
2028:=
2020:j
2017:i
2007:N
2002:1
1999:=
1996:j
1987:1
1982:=
1977:i
1968:x
1927:.
1922:]
1916:1
1909:1
1896:0
1891:0
1884:1
1870:1
1860:1
1853:1
1840:0
1835:0
1828:0
1821:1
1811:1
1804:1
1791:0
1784:0
1779:0
1772:1
1762:1
1755:1
1738:1
1725:0
1720:0
1713:1
1703:1
1697:[
1692:=
1687:j
1684:i
1668:1
1665:α
1656:α
1639:B
1635:A
1631:C
1627:C
1623:B
1619:B
1615:A
1599:.
1597:B
1593:C
1589:A
1585:C
1581:B
1577:B
1573:A
1540:i
1529:2
1513:.
1508:]
1475:[
1470:=
1465:]
1459:1
1452:1
1445:1
1438:1
1432:[
1425:1
1417:)
1411:(
1406:=
1398:x
1341:i
1321:1
1318:=
1313:i
1309:K
1284:]
1278:1
1251:1
1246:0
1224:1
1219:0
1212:0
1197:1
1191:[
1186:=
1177:]
1150:1
1144:[
1139:=
1136:r
1115:4
1112:x
1087:α
1082:N
1058:}
1054:1
1051:=
1046:i
1042:x
1036:i
1028:,
1025:0
1017:i
1013:x
1009::
1004:i
1000:x
995:{
991:=
986:1
980:N
962:N
954:C
943:N
939:N
931:N
927:n
923:n
919:N
911:N
902:.
894:n
877:N
870:i
863:i
861:x
851:i
844:i
842:r
837:j
833:i
824:α
807:α
802:N
787:)
781:j
777:x
771:j
768:i
758:N
753:1
750:=
747:j
736:1
732:(
726:i
722:x
716:i
712:r
708:=
702:t
699:d
692:i
688:x
684:d
660:)
652:i
648:K
641:j
637:x
631:j
628:i
618:N
613:1
610:=
607:j
593:1
589:(
583:i
579:x
573:i
569:r
565:=
559:t
556:d
549:i
545:x
541:d
528:i
520:α
508:N
497:α
488:α
479:α
458:.
454:)
449:)
442:2
438:K
431:1
427:x
413:+
408:2
404:x
397:(
390:1
386:(
380:2
376:x
370:2
366:r
362:=
352:t
349:d
342:2
338:x
334:d
323:)
318:)
311:1
307:K
300:2
296:x
282:+
277:1
273:x
266:(
259:1
255:(
249:1
245:x
239:1
235:r
231:=
221:t
218:d
211:1
207:x
203:d
185:2
182:x
176:1
173:x
156:K
152:r
148:x
132:.
128:)
122:K
119:x
111:1
107:(
103:x
100:r
97:=
91:t
88:d
83:x
80:d
23:.
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