Mathematics and Biology

More realistic models of exploitation
The dynamics of biological exploitation.
Math Awareness Month 1999

More realistic models of exploitation

he Lotka-Volterra equations make two particularly unreasonable assumptions. 
    1. The equations postulate that, in the absence of predators, the victim population grows indefinitely.  This is a consequence of the fact that, in the Lotka-Volterra equations, predation is the sole limitation on the growth rate of the victim population.
    2. The Lotka-Volterra equations assume that predators are insatiable.  Thus, if the density of prey doubles, the predators eat twice as much; they never get full!
We can relax the first of these assumptions by making the assumption that competition between prey for some resource (such as food or space) limits their densities in the absence of predation.  This gives us the equations
MacArthur's consumer-resource equations.

The new parameter  expresses the effects of intraspecific competition.

Phase portrait and time series plots for predator-prey dynamics with finite victim carrying capacity.As we see from the phase plane picture and from the time series plots, the dynamics are now stable.  That is, the predator-prey oscillations are now damped.

What happens when we relax the second unrealistic assumption?

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