**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

II. Changing the assumptions of the initial model to make it more realistic

A. Relax Assumption 1:

-Prey controlled by predator
and
by other density-dependent factors (like limits on food resources)

-Implications: Stabilizes
interactions (no oscillations)

B. Relax Assumption 1:

- In addition to above, prey
population growth limited at low prey population sizes

– Might be hard to find
mates

- Implications: Stability
depends
on predator efficiency. If they are efficient, then
predators
will hunt prey to extinction.

C. Relax assumption 3:

- Predators are not able to
consume infinite prey

-Functional response of
predators
to prey density

- Implications: Predators
that
switch prey or that reach carrying capacity stabilize
predator-prey
interactions

- Final note: if realistic
assumptions tend to stabilize dynamics, what causes
Oscillations?

- Time delays