Discrete Population Model for Fishery Stocks

bifurcation diagram

population density function

intrinsic growth rate r

3.1024

This Demonstration displays the bifurcation diagram for a realistic population dynamic model given by

x

i+1

=

x

i

exp[r(1-

x

i

/k)]

, where

r

is the intrinsic growth rate (taken here as a bifurcation parameter),

x

i

is the number of fish (or density of population) at generation

i

, and

k

is the population capacity of the environment, set equal to 1 here. This mathematical expression was given by W. E. Ricker (1954), who invented a discrete population model for fishery stocks. The model can be used to predict the number of fish in a fishery. When the cycle of period three appears,

r=3.1024

. As expected, for higher values of

r

, we observe chaotic behavior. Snapshots 2 to 5 present period three

(r=3.1024)

, period two

(r=2.4444)

, period four

(r=2.5873)

, and chaotic behaviors

(r=3.3175)

, respectively.