Sim.DiffProc (version 2.5)

SLVM: Stochastic Lotka-Volterra Model

Description

Simulation the stochastic Lotka-Volterra model.

Usage

SLVM(N, t0, T, x0, y0, a, b, c, d, sigma, Step = FALSE, Output = FALSE)

Arguments

N

size of process.

t0

initial time.

T

final time.

x0

initial value of the process at time t0 (x0 > 0).

y0

initial value of the process at time t0 (y0 > 0).

a

positive parameter.

b

positive parameter.

c

positive parameter.

d

positive parameter.

sigma

positive parameter.

Step

if Step = TRUE ploting step by step.

Output

if output = TRUE write a output to an Excel (.csv).

Value

• data.frame(time,x,y), plot 1D and 2D of the process.

Details

The Lotka-Volterra system of stochastics differential equations, (Lotka (1925),Volterra (1926)): $$dX(t) = (a * X(t) - b * X(t) * Y(t)) dt + sigma* dW1(t)$$ $$dY(t) = (c * X(t) * Y(t) - d * Y(t)) dt + sigma* dW2(t)$$ with positive x0, y0 and positive parameters a, b, c, d describes a behaviour of a prey-predator system in terms of the prey and predator (intensities) X(t) and Y(t). Here, a is the rate of increase of prey in the absence of predator, d is a rate of decrease of predator in the absence of prey while the rate of decrease in prey is proportional to the number of predators b* Y(t), and similarly the rate of increase in predator is proportional to the number of prey c* X(t). The system possesses the first integral which is a closed orbit in the first quadrant of phase plane x, y. It is given by : $$r(x,y) = c * x - d * log(x) + b * y - a * log(y) + r0$$

References

Fima C Klebaner. Introduction to stochastic calculus with application (Second Edition), Imperial College Press (ICP), 2005.

See Also

WFD Feller Branching Diffusion, FBD Feller Branching Diffusion.

Examples

SLVM(N=5000,t0=0,T=100,x0=1,y0=1,a=1,b=2,c=0.5,d=0.25,sigma=0.01)