On a stochastic differential equation modeling of prey-predator evolution

Abstract

We study a stochastic differential equation model of prey-predator evolution. To keep in line with a known deterministic model we include the social and interaction terms with the drift in our model, and the randomness arises as fluctuations in the ecosystem. The notion of equilibrium population level and special types of fluctuations force us to work with degenerate elliptic operators. We consider the propagation of the population system in a sufficiently large but bounded domain. This enables us to look at not only the population extinction, but also the explosion beyond a certain level. Both extinction and explosion are possible; and, when they are not, we show that the population asymptotically reaches the equilibrium level. We show that the extinction, explosion and saturation probabilities satisfy, as functions of the initial population size, an integral equation arising out of a Dirichlet problem for a non-degenerate elliptic equation; and these probabilities are also smooth solutions of the Dirichlet problems. They are also used to express the solution of another Dirichlet problem for a degenerate elliptic equation.

A stochastic competing-species model and ergodicity

“…Gard and Kannan [6] studied the classical prey-predator model with the rates perturbed by Gaussian white noise. They studied (3) in the case that γ < 0 and δ < 0, and showed that, on the set of realizations where it remains bounded for all time, the solution converges to the fixed point (x * , y * ) [6, Theorem 3.1] or converges to the boundary of the bounding set [6, Theorem 3.2].…”

“…Recall the definition of F in (6). From (7), we have Stochastic competing species and ergodicity 747 and, therefore,…”

“…Consider the functions F , given by (6), and H (u) = e cu , where c is a positive constant to be determined later. Notice that H is nonnegative and monotone increasing.…”

Likelihood ratio test for a special predator-prey system

“…In this model, the error process is considered as mainly measurement (observation) error, which does not interfere with the deterministic function. In practice, this assumption is related to the fact that, had this interference been present, the oscillatory behaviour might have been lost, as is the case in the corresponding Lotka-Volterra stochastic differential equations (SDEs) models (see, e.g., [7][8][9]). …”

Stochastic three species systems

“…Tsokos and Hinkley (1973) studied the behaviour of the system by formulating a general bivariate stochastic model and derived estimators for its parameters. Gard and Kannan (1976) developed a stochastic differential equation model of prey-predator evolution and studied population extinction. Billard (1977) considered the case in which interactions of two species occur over a sufficiently short period of time during which no births occur, and provided complete solutions for this stochastic model.…”