**Lecture 23: Predator-prey theory**

I. Predator-prey oscillations: a model

A. The Lotka-Volterra model of predator and prey population dynamics (p. 325-328 in your book)

B. Two equations, one for predator (Population size denoted as P for predator instead of N) and one for prey (N)

C. Without the predator, assume
exponential population growth of prey, dN/dt = rN

1. Assumption 1:
prey population growth is only limited by predators

2. Assumption
2: predator is a specialist, consuming only this prey

3. Assumption 3:
Predator population can consume an infinite number of prey

4. With the predator,
assume that population is reduced by the predator in proportion to
the predator's capture rate, c

-- thus, dN/dt = rN - cNP

D. Without prey, assume that predator
population sizes decline exponentially, dP/dt = -dP

1. d is the death
rate

2. In the
presence of the prey, assume that predator population growth
depends on the capture rate of prey, and on the rate at which
predators convert prey to energy

-- thus, dP/dt = acNP - dP

E. Use a graphical approach to solving for the dynamics of predator and prey population sizes

1. Solve
Lotka-Volterra equations – one species at a time for
solutions at equilibrium (when population size is not changing)

2. Graph the
line represented by the equilibrium solutions

3. Determine
population growth from any initial population size