Class 2: Writing and Analyzing Population and Predator-Prey Models

(Book sections 1.4)

Simple Population Model

X’(t) = bX(t) - dX(t) = (b-d)X(t) = rX(t)

• EXPONENTIAl growth

• not realistic in the long run

• assumption: unlimited resources

Population Model with Carrying Capacity

X(t) / k = the fraction of resources used

1 - (X(t) / k) = the fraction of resources left

THEREFORE

X’(t) = rX(t) (1 - (X(t) / k)) 

• LOGISTIC model!!

Lotka-Volterra Model (aka Shark-Tuna Model)

T(t) = Prey

S(t) = Predator

T’(t) = bT(t) - β S(t) T(t)

S’(t) = mβS(t) T(t) - dS(t)

state variables: S(t), T(t)
parameters: β, m, b, d

lit

lit