Editing Competitive Lotka–Volterra equations (section)

==Overview==
The form is similar to the [[Lotka–Volterra equations]] 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 [[exponential function|exponential]].  For the competition equations, the [[Logistic function#Logistic differential equation|logistic equation]] is the basis.

The logistic population model, when used by [[ecology|ecologists]] often takes the following form:
<math display="block">{dx \over dt} = rx\left(1-{x \over K}\right).</math>

Here {{mvar|x}} is the size of the population at a given time, {{mvar|r}} is inherent per-capita growth rate, and {{mvar|K}} is the [[carrying capacity]].

===Two species===
Given two populations, {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}}, with logistic dynamics, the Lotka–Volterra formulation adds an additional term to account for the species' interactions.  Thus the competitive Lotka–Volterra equations are:
<math display="block">\begin{align}
{dx_1 \over dt} &= r_1 x_1\left(1-\left({x_1+\alpha_{12}x_2 \over K_1}\right) \right) \\[0.5ex]
{dx_2 \over dt} &= r_2 x_2\left(1-\left({x_2+\alpha_{21}x_1 \over K_2}\right) \right).
\end{align}</math>

Here, {{math|''α''<sub>12</sub>}} represents the effect species 2 has on the population of species 1 and {{math|''α''<sub>21</sub>}} 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 ''α''-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,<ref>{{cite journal | last=Bomze | first=Immanuel M. |author-link=Immanuel Bomze| title=Lotka-Volterra equation and replicator dynamics: A two-dimensional classification | journal=Biological Cybernetics | publisher=Springer Science and Business Media LLC | volume=48 | issue=3 | year=1983 | issn=0340-1200 | doi=10.1007/bf00318088 | pages=201–211| s2cid=206774680 }}</ref><ref>{{cite journal | last=Bomze | first=Immanuel M. |author-link=Immanuel Bomze| title=Lotka-Volterra equation and replicator dynamics: new issues in classification | journal=Biological Cybernetics | publisher=Springer Science and Business Media LLC | volume=72 | issue=5 | year=1995 | issn=0340-1200 | doi=10.1007/bf00201420 | pages=447–453| s2cid=18754189 }}</ref> which is based upon equivalence to the 3-type [[replicator equation]].

===''N'' species===
This model can be generalized to any number of species competing against each other.  One can think of the populations and growth rates as [[vector (geometric)|vectors]], {{mvar|α}}'s as a [[matrix (mathematics)|matrix]].  Then the equation for any species {{mvar|i}} becomes
<math display="block">\frac{dx_i}{dt} = r_i x_i \left(1- \frac{\sum_{j=1}^N \alpha_{ij}x_j}{K_i} \right) </math>
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),
<math display="block">\frac{dx_i}{dt} = r_i x_i \left( 1 - \sum_{j=1}^N \alpha_{ij}x_j \right) </math>
where {{mvar|N}} is the total number of interacting species.  For simplicity all self-interacting terms {{math|''α''<sub>ii</sub>}} are often set to 1.

===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.