Regularity of Solutions to the Dirichlet Problem for Degenerate Elliptic Equation

2003 ◽  
Vol 24 (04) ◽  
pp. 529-540 ◽  
Author(s):  
Yemin Chen
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Degenerate Elliptic Equation ◽  
Regularity Of Solutions ◽  
Degenerate Elliptic

On a stochastic differential equation modeling of prey-predator evolution

1976 ◽  
Vol 13 (3) ◽  
pp. 429-443 ◽  
Author(s):  
T. C. Gard ◽  
D. Kannan
Keyword(s):  
Differential Equation ◽  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Stochastic Differential Equation ◽  
Deterministic Model ◽  
Degenerate Elliptic Equation ◽  
Equation Modeling ◽  
Initial Population ◽  
Dirichlet Problems ◽  
Degenerate Elliptic

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.

On a stochastic differential equation modeling of prey-predator evolution

1976 ◽  
Vol 13 (03) ◽  
pp. 429-443 ◽  
Author(s):  
T. C. Gard ◽  
D. Kannan
Keyword(s):  
Differential Equation ◽  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Stochastic Differential Equation ◽  
Deterministic Model ◽  
Degenerate Elliptic Equation ◽  
Equation Modeling ◽  
Initial Population ◽  
Dirichlet Problems ◽  
Degenerate Elliptic

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.

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