Symplectic methods applied to the Lotka-Volterra system

• Main Author: Beck, Mélanie
• Format: Others
• Language: en
• Published: McGill University 2003
• Subjects: Pure Sciences - Mathematics
• Online Access: http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=19583

We analyse the preservation of physical properties of numerical approximations tosolutions of the Lotka-Volterra system: its positivity and the conservation of theHamiltonian. We focus on two numerical methods : the symplectic Euler method andan explicit variant of it. We first state under which conditions they are symplectic andwe prove they are both Poisson integrators for the Lotka-Volterra system. Then, westudy under which conditions they stay positive. For the symplectic Euler method,we derive a simple condition under which the numerical approximation always stayspositive. For the explicit variant, there is no such simple condition. Using propertiesof Poisson integrators and backward error analysis, we prove that for initial conditionsin a given set in the positive quadrant, there exists a bound on the step size, such thatnumerical approximations with step sizes smaller than the bound stay positive overexponentially long time intervals. We also show how this bound can be estimated.We illustrate all our results by numerical experiments.