## The Lotka Volterra Competition Model

Formulated independently by A. J. Lotka in 1925 and V. V. Volterra in 1926, this is the oldest and the simplest demographic model of competition. Two or more populations compete in a limiting habitat in which a maximum number of individuals of each species is allowed. The classical Lotka-Volterra set of equations is based on the logistic growth of populations and incorporates explicitly both intra- and interspecific growth limitation. It is an extension of the Verhulst's logistic equation, which only included intraspecific competition as a density-dependent growth limitation.

Each population i is represented by its size or density Ni and by two species-specific demographic parameters: an intrinsic potential per capita growth rate ri and a constant carrying capacity K, that is the maximum possible density it can reach in the habitat. Direct asymmetric interactions between pairs of populations are represented by competition coefficients a.j The competition coefficient ai:i is a parameter quantifying the negative effect of an individual of species j on the exponential growth rate of the population of species i. This is the simplest way to represent interference competition among populations. All competition coefficients build a square interaction matrix with all positive terms. In this interaction matrix, all diagonal terms are equal to 1 and represent intraspecific competition. For every species i, the kinetics of the population density is described by the following ordinary differential equation:

in which the red part represents the reduction in exponential growth rate due to the overall competition effect in the plant community in regard to the carrying capacity for the species.

For two competing populations of species 1 and 2, the model is made of two equations, one for each population:

dNi Ni aiN

In each equation, the blue part represents the negative effect of intraspecific competition (inherited from the logistic equation) and the red part the negative effect of interspecific competition on the exponential growth rate of each population.

Although this model has no closed-form analytical solution, a graphical stability analysis shows that, in the long term, the coexistence of the two species depends on two main parameter configurations (Figure 1). The phase space is the plane of the two state variables N1 and N2. For each species, we can draw a 'zero net growth isocline' (ZNGI), representing all couples of values for which its population density does not change. These isoclines separate regions ofpositive (for lower densities) and negative (for higher densities) net growth rates for each population. Note that trajectories cross species-1's ZNGI vertically and species-2's ZNGI horizontally. We can identify four possible conditions for this system to be in equilibrium, that is, with both net growth rates equal to zero and therefore constant equilibrium values N and N2:

1. N[ = 0 and N2 = 0, both populations equal to 0 (trivial case);

2. N = K1 and N, = 0, species 1 outcompetes species 2;

3. N, = 0 and N, = K2, species 2 outcompetes species 1;

4. N, > 0 and N, >0, species 1 and 2 coexist at the intersection of the ZNGIs.

The coexistence condition at equilibrium, with both N > 0 and N >0, is that the two isoclines cross in the positive orthant of the phase space. The stability condition for this coexistence equilibrium is that < In this stable case (Figure 1a), for which a12 < K1/K2 and a21 < K2/Kb intraspecific competition is greater than interspecific competition and the intersection point of the isoclines is the only attractor of the system. Otherwise, in the unstable case (Figure 1b), if both a12 > Ki/K2 and a21 > K2/Kb then interspecific competition is greater than intraspecific competition, preventing coexistence to recover after any perturbation: the intersection point is an unstable equilibrium at the boundary of two basins of attraction leading each to a different attrac-tor, with either species 1 or species 2 outcompeting the other one. This phenomenon, by which eventually one species will exclude the other, but the winner depends on the initial abundances of the two species, is called 'founder control'.

The main assumptions of the Lotka-Volterra competition model are: (1) implicit renewable resources are limiting when the population density approaches its carrying capacity; (2) competition coefficients and carrying capacities are fixed attributes of the species; and (3) density-dependence, that is, per capita effects of density on realized per capita growth rate, is linear for both intra- and interspecific competition (linear isoclines). Since it is based on the logistic growth of populations, it also assumes that populations have no age or genetic structure, that there is no migration and no time lags.

This phenomenological model based on the 'logistic-competition theory' was severely criticized, particularly in respect to the lack of realism of the competition coefficient and because it does not explicitly consider changes

Stable coexistence

Unstable coexistence Unstable coexistence Figure 1 Trajectories and equilibrium states of the two-species Lotka-Volterra model in the phase space. Green stars represent stable equilibrium points (attractors), whereas yellow stars represent unstable equilibrium points (repellors). Colored straight lines are zero net growth isoclines of each species (red: species 1; blue: species 2). Dotted lines are examples of trajectories starting from various initial conditions for population densities. In case a, trajectories converge to the intersection point of the two isoclines allowing the stable coexistence of the two species. In case b, trajectories diverge to end up with the survival of one of the two species only, depending on the initial conditions. The dashed line is the separatrix between the two basins of attraction.

in resources utilized by the competitors. Thereafter, models of plant competition aimed at providing a more mechanistic representation of the process. 