Numerical Methods for Solving the Lotka-Volterra Equations
Introduction
The Lotka-Volterra system arises in mathematical biology and models the growth of animal species. Consider two species where
denotes the number of predators and
denotes the number of prey. A particular case of the Lotka-Volterra differential system is
where the dot denotes differentiation with respect to time
.
The Lotka-Volterra system (
1) has an invariant
, which is constant for all
:
The level curves of the invariant (
2) are closed so that the solution is periodic. It is desirable that the numerical solution of (
3) is also periodic, but this is not always the case. Note that (
4) is a Poisson system:
where
is defined in (
5).
Poisson systems and Poisson integrators are discussed in Chapter VII.2 of [
HLW02] and [
MQ02].
Load a package with some predefined problems and select the Lotka-Volterra system.
Define a utility function for visualizing solutions.
Explicit Euler
Use the explicit or forward Euler method to solve the system (
6).
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Backward Euler
Define the backward or implicit Euler method in terms of the RadauIIA implicit Runge-Kutta method and use it to solve (
7). The resulting trajectory spirals from the initial conditions toward a fixed point at
in a clockwise direction.
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Projection
Projection of the forward Euler method using the invariant (
8) of the Lotka-Volterra equations gives a periodic solution.
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Splitting
Another approach for obtaining the correct qualitative behavior is to additively split (
9) into two systems:
By appropriately solving (
10) it is possible to construct Poisson integrators.
Define the equations for splitting of the Lotka-Volterra equations.
Symplectic Euler
Define the symplectic Euler method in terms of a splitting method using the backward and forward Euler methods for each system in (
11).
The numerical solution using the symplectic Euler method is periodic.
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Flows
Consider splitting the Lotka-Volterra equations and computing the flow (or exact solution) of each system in (
12). The solutions can be found as follows, where the constants should be related to the initial conditions at each step.
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An advantage of locally computing the flow is that it yields an
explicit, and hence very efficient, integration procedure. The
method provides a general way of computing the flow of each splitting using
DSolve only during the initialization phase.
Set up a hybrid symbolic-numeric splitting method and use it to solve the Lotka-Volterra system.
The numerical solution using the splitting method is periodic.
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