### LOTKA-VOLTERA EQUATION (LVE)

**9. LOTKA-VOLTERRA EQUATION (LVE)**

Lotka-Volterra equation describes an ecological predator-prey model which assumes a set of fixed positive constants. A (the growth rate of prey), B (the rate at which predators destroy prey), C (the death rate of predators), and D (the rate at which predators increase by consuming prey).

The following conditions were given in the equation.

(1) A prey population x increases at a rate dx = Axdt (proportional to the number of prey)but simultaneously preys are destroyed by the predators at a rate dx = Bxydt (proportional to the numbers of prey and predators).

(2) A predator population y decreases at a rate dy = Cydt (proportional to the number of predators), but increases at a rate dy = Dx ydt (again proportional to the product of the numbers of prey and predators).

**Note:** Prey are showed in bold in the graphs and predators are unbold. Thus the prey curve always leads the predator curve.

**This gives two coupled differential equations:-**

Critical points are attained when

dx/dt = dy/dt = O, so

A – By = 0

– C + Dn = 0

The sole stationary point is therefore located at (x, y) = (C/D, A/B).

**Interspecific Relation and Lotka Volterra Curves:-**

When a species of prey increases the number of the individuals of predator species also increases. When predators increase drastically in number then prey starts to become extinct fast.

If prey decreases then predators also start to decrease so finally they make an equilibrium in their population size or number as shown in the graph.

Lotka-Volterra model contains the relationship between two species, one is the predator and another is prey. It involves differential equations of the dynamics of both populations (Predator-prey, parasite-host or herbivore-plant). It was developed independently by Alfred Lotka and Vito Volterra.

The model simplifies several assumptions. In those assumptions, some are given below:-

(1) The prey population will grow exponentially when the predator is absent.

(2) The predator population decreases in the absence of prey's population.

(3) Both of the species make an equilibrium to maintain their population size.

(4) There is no environment complexity i.e. both populations are moving randomly through a homogenous environment.

(5) In Lotka Volterra equation, Intrinsic growth rate (r), competition coefficient (d) and carrying (k) all are constants.

(6) Every individual within each population is identical.

(7) The population is not allowed to diversity.

(8) The environment (habitats) is homogenous.

Lotka Volterra Equation is written as follows for population first.

r = intrinsic rate constant for population first

N_{1} = Initial number of the population one

N_{2} = Initial number of population second

Alpha = Inhibitory effect of species second on species first (parasite on the host)

**The equation for population 2**

Beta = Inhibitory effect of species first on species second

By Lotka Volterra equations it is predicted that both competing species will co-exist or one of them eliminated or not eliminated. So the total of four cases is possible.

Case 1:- Species two will be eliminated

In the above figure, two isoclines do not cross and isocline of the population. N_{1} is above than N_{2} any point located below N_{2} isocline represents co-existence of both populations of species and indicates for the increase of both populations. Any point located above N1 isocline represents both species are decreasing. So K_{1}/alpha is more than K_{2}. Hence a will be less. K_{2}/beta will be less than K_{1} sob will be more so species second will be affected more from species first. Hence species second will be eliminated.

Beta is more = so, Inhibitory effect of species first on species second will be more and it will be eliminated.

Case - 2:– Species first will be eliminated

In the above figure, two isoclines do not cross and isocline of population N_{2} is above than N_{1} any point located below N_{1} isocline represents co-existence of both population of species and indicates for the increase of both population. Any point located above N_{2} isocline represents both species are decreasing. So, K_{2} is more than K1/a. Hence a will be more. K_{1} will be less than K_{2}/beta so a will be more, so species first will be affected more from species second. Hence, species first will be eliminated.

Case–3:- Either of the species can be eliminated.

In the above figure, two isoclines cross each other. In one isocline, K_{2} is more than K_{1}/alpha. So a is more than b hence, K_{1} will be less so species one can be eliminated. In another isocline, K_{1} is more than K_{2}/beta. Beta is also more, so K_{2} will be more. Hence, species two can also be eliminated.

Alpha is more = so, Inhibitory effect of species second on species first will be more so, species first can be eliminated.

Beta is more = so, Inhibitory effect of species first on species second will be more so, species second can be eliminated.

Case–4:- Both Species coexist.

In the above figure, two isoclines cross each other. In one isocline, K_{1} is more than K_{2}/b. b is zero here so K_{2} is more than K_{1}. In the second isocline, K_{1}/a is more than K_{2}. a is zero here so K_{1} is more than K_{2} here. Hence a and b both are affected. So both species can exist.

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