Editing Competitive Lotka–Volterra equations (section)

===Possible dynamics===
The definition of a competitive Lotka–Volterra system assumes that all values in the interaction matrix are positive or 0 ({{math|''α<sub>ij</sub>'' ≥ 0}} for all {{mvar|i}}, {{mvar|j}}).  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 ({{math|''r<sub>i</sub>'' > 0}} for all {{mvar|i}}), then some definite statements can be made about the behavior of the system.

# The populations of all species will be bounded between 0 and 1 at all times ({{math|0 ≤ ''x<sub>i</sub>'' ≤ 1}}, for all {{mvar|i}}) as long as the populations started out positive.
# Smale<ref>{{cite journal | last=Smale | first=S. | title=On the differential equations of species in competition | journal=Journal of Mathematical Biology | publisher=Springer Science and Business Media LLC | volume=3 | issue=1 | year=1976 | issn=0303-6812 | doi=10.1007/bf00307854 | pages=5–7| pmid=1022822 | s2cid=33201460 }}</ref> showed that Lotka–Volterra systems that meet the above conditions and have five or more species (''N'' ≥ 5) can exhibit any [[Asymptote|asymptotic]] behavior, including a [[Fixed point (mathematics)|fixed point]], a [[limit cycle]], an [[Torus|''n''-torus]], or [[Chaotic attractor|attractors]].
# Hirsch<ref>{{cite journal | last=Hirsch | first=Morris W. | title=Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere | journal=SIAM Journal on Mathematical Analysis | publisher=Society for Industrial & Applied Mathematics (SIAM) | volume=16 | issue=3 | year=1985 | issn=0036-1410 | doi=10.1137/0516030 | pages=423–439| url=https://escholarship.org/uc/item/67z7c17v }}</ref><ref>{{cite journal | last=Hirsch | first=M W | title=Systems of differential equations which are competitive or cooperative: III. Competing species | journal=Nonlinearity | publisher=IOP Publishing | volume=1 | issue=1 | date=1988-02-01 | issn=0951-7715 | doi=10.1088/0951-7715/1/1/003 | pages=51–71| bibcode=1988Nonli...1...51H | s2cid=250848783 | url=https://escholarship.org/uc/item/9w89b10z }}</ref><ref>{{cite journal | last=Hirsch | first=Morris W. | title=Systems of Differential Equations That are Competitive or Cooperative. IV: Structural Stability in Three-Dimensional Systems | journal=SIAM Journal on Mathematical Analysis | publisher=Society for Industrial & Applied Mathematics (SIAM) | volume=21 | issue=5 | year=1990 | issn=0036-1410 | doi=10.1137/0521067 | pages=1225–1234}}</ref> proved that all of the dynamics of the attractor occur on a [[manifold]] of dimension ''N''−1.  This essentially says that the attractor cannot have [[dimension]] greater than ''N''−1.  This is important because a limit cycle cannot exist in fewer than two dimensions, an ''n''-torus cannot exist in less than ''n'' dimensions, and chaos cannot occur in less than three dimensions.  So, Hirsch proved that competitive Lotka–Volterra systems cannot exhibit a limit cycle for ''N'' < 3, or any [[torus]] or chaos for ''N'' < 4.  This is still in agreement with Smale that any dynamics can occur for ''N'' ≥ 5.
#*More specifically, Hirsch showed there is an [[Invariant (mathematics)|invariant]] set ''C'' that is [[Homeomorphism|homeomorphic]] to the (''N''−1)-dimensional [[simplex]] <math display="block">\Delta_{N-1} = \left \{ x_i : x_i \ge 0, \sum_i x_i = 1 \right \}</math> and is a global attractor of every point excluding the origin.  This carrying simplex contains all of the asymptotic dynamics of the system.
# To create a stable ecosystem the α<sub>ij</sub> matrix must have all positive eigenvalues.  For large-''N'' systems Lotka–Volterra models are either unstable or have low connectivity.  Kondoh<ref>{{cite journal | last=Kondoh | first=M. | title=Foraging Adaptation and the Relationship Between Food-Web Complexity and Stability | journal=Science | publisher=American Association for the Advancement of Science (AAAS) | volume=299 | issue=5611 | date=2003-02-28 | issn=0036-8075 | doi=10.1126/science.1079154 | pages=1388–1391| pmid=12610303 | s2cid=129162096 }}</ref> and Ackland and Gallagher<ref>{{cite journal | last1=Ackland | first1=G. J. | last2=Gallagher | first2=I. D. | title=Stabilization of Large Generalized Lotka-Volterra Foodwebs By Evolutionary Feedback | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=93 | issue=15 | date=2004-10-08 | issn=0031-9007 | doi=10.1103/physrevlett.93.158701 | page=158701| pmid=15524949 | bibcode=2004PhRvL..93o8701A }}</ref> have independently shown that large, stable Lotka–Volterra systems arise if the elements of {{mvar|α<sub>ij</sub>}} (i.e. the features of the species) can evolve in accordance with natural selection.