10.2 Lotka-Volterra ModelLotka-Volterra model is the simplest model of predator-prey interactions. The model was developed independently by Lotka (1925) and Volterra (1926):
It has two variables (P, H) and several parameters:
Measuring parameters of the Lotka-Volterra modelThe following set of experiments should be done:
How to solve differential equationsThere are two major approaches: analytical and numerical. Analytical methods are complicated and require good mathematical skills. Also, many differential equations have no analytical solution at all. Numerical methods are easy and more universal (however, there are problems with convergence).The simplest and least accurate is the Euler's method. Consider a stationary differential equation: First we need initial conditions. We will assume that at time to the function value is x(to). Now we can estimate x-values at later (or earlier) time using equation:
The main source of error in the Euler's method is estimation of derivative at the start of time interval. The direction of actual solution may change drastically during this time interval and numerically predicted point could be far from the actual solution (see the figure). Euler's method can be improved, if the derivative (slope) is estimated at the center of time interval . However, the derivative at the center depends on the function value at the center which is unknown. Thus, first we need to estimate the function value at the middle point using simple Euler's method, and then we can estimate the derivative at the middle point.
k is the function value in the center of time interval l. Finally, we can estimate function value at the end of the time interval:
This method is applied to Lotka-Volterra equations in the following Excel spreadsheet: . Excel spreadsheet "lotka.xls" First, we estimate prey and predator densities (H' and P', respectively) at the center of time interval:
The second step is to estimate prey and predator densities (H" and P" at the end of time step l:
These two graphs were plotted using the same model parameters. The only difference is in initial density of prey. This model has no asymptotic stability, it does not converge to an attractor (does not "forget" initial conditions).
This figure shows relative changes in prey predator density for both initial conditions. Trajectories are closed lines. The model of Lotka and Volterra is not very realistic. It does not consider any competition among prey or predators. As a result, prey population may grow infinitely without any resource limits. Predators have no saturation: their consumption rate is unlimited. The rate of prey consumption is proportional to prey density. Thus, it is not surprising that model behavior is unnatural showing no asymptotic stability. However numerous modifications of this model exist which make it more realistic. Additional information on the Lotka-Volterra model can be found at other WWW sites:
References:Lotka, A. J. 1925. Elements of physical biology. Baltimore: Williams & Wilkins Co.Volterra, V. 1926. Variazioni e fluttuazioni del numero d'individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei. Ser. VI, vol. 2. |