[1][2] They can be used to model direct competition and trophic relationships between an arbitrary number of species.
This makes them useful as a theoretical tool for modeling food webs.
However, they lack features of other ecological models such as predator preference and nonlinear functional responses, and they cannot be used to model mutualism without allowing indefinite population growth.
The generalised Lotka-Volterra equations model the dynamics of the populations
They are a set of ordinary differential equations given by where the vector
[3] The generalised Lotka-Volterra equations can represent competition and predation, depending on the values of the parameters, as described below.
"Generalized" means that all the combinations of pairs of signs for both species (−/−,−/+,+/-, +/+) are possible.
are the intrinsic birth or death rates of the species.
means that species i is able to reproduce in the absence of any other species (for instance, because it is a plant that is wind pollinated), whereas a negative value means that its population will decline unless the appropriate other species are present (e.g. a herbivore that cannot survive without plants to eat, or a predator that cannot persist without its prey).
The effect is proportional to the populations of both species, as well as to the value of
However, this is not often used in practice, because it can make it possible for both species' populations to grow indefinitely.
For example, if two predators eat the same prey then they compete indirectly, even though they might not have a direct competition term in the community matrix.
This self-limitation prevents populations from growing indefinitely.
The generalised Lotka-Volterra equations are capable of a wide variety of dynamics, including limit cycles and chaos as well as point attractors (see Hofbauer and Sigmund[2]).
If the fixed point is unstable then there may or may not be a periodic or chaotic attractor for which all the populations remain positive.
species, a complete classification of this dynamics, for all sign patterns of above coefficients, is available,[4] which is based upon equivalence to the 3-type replicator equation.
In the case of a single trophic community, the trophic level below the one of the community (e.g. plants for a community of herbivore species), corresponding to the food required for individuals of a species i to thrive, is modeled through a parameter Ki known as the carrying capacity.
E.g. suppose a mixture of crops involving S species.
can be thus written in terms of a non-dimensional interaction coefficient
A straightforward procedure to get the set of model parameters
is to perform, until the equilibrium state is attained: a) the S single species or monoculture experiments, and from each of them to estimate the carrying capacities as the yield of the species i in monoculture
(the superscript ‘ex’ is to emphasize that this is an experimentally measured quantity a); b) the S´(S-1)/2 pairwise experiments producing the biculture yields,
Using this procedure it was observed that the Generalized Lotka–Volterra equations can predict with reasonable accuracy most of the species yields in mixtures of S >2 species for the majority of a set of 33 experimental treatments acrossdifferent taxa (algae, plants, protozoa, etc.).
[6] The vulnerability of species richness to several factors like, climate change, habitat fragmentation, resource exploitation, etc., poses a challenge to conservation biologists and agencies working to sustain the ecosystem services.
Hence, there is a clear need for early warning indicators of species loss generated from empirical data.
A recently proposed early warning indicator of such population crashes uses effective estimation of the Lotka-Volterra interaction coefficients
The idea is that such coefficients can be obtained from spatial distributions of individuals of the different species through Maximum Entropy.
This method was tested against the data collected for trees by the Barro Colorado Island Research Station, comprising eight censuses performed every 5 years from 1981 to 2015.
is always steeper and occurs before the drop of the corresponding species abundance Ni .
occur between 5 and 15 years in advance than comparable declines for Ni, and thus they serve as early warnings of impending population busts.