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alt="sigma"/> have the form:

mu left-parenthesis t comma r right-parenthesis equals alpha left-parenthesis t right-parenthesis r plus beta left-parenthesis t right-parenthesis comma sigma left-parenthesis t comma r right-parenthesis equals StartRoot gamma left-parenthesis t right-parenthesis r plus delta left-parenthesis t right-parenthesis EndRoot

      where alpha left-parenthesis t right-parenthesis comma beta left-parenthesis t right-parenthesis comma gamma left-parenthesis t right-parenthesis and delta left-parenthesis t right-parenthesis are given functions of t.

      The coefficients A(t, T) and B(t, T) in this case are determined by the following ordinary differential equations:

      and:

      3.3.3 Predator-Prey Models

StartFraction italic d upper P Over italic d t EndFraction equals italic r upper P left-parenthesis 1 minus StartFraction upper P Over upper K EndFraction right-parenthesis comma upper P left-parenthesis 0 right-parenthesis equals upper P 0

      where r is the growth rate and K is the carrying capacity. The initial population is upper P 0. It is easy to check the following identities:

upper P left-parenthesis t right-parenthesis equals StartFraction italic upper K upper P 0 e Superscript italic r t Baseline Over upper K plus upper P 0 left-parenthesis e Superscript italic r t Baseline minus 1 right-parenthesis EndFraction and limit Underscript t right-arrow infinity Endscripts upper P left-parenthesis t right-parenthesis equals upper K period

      Transformation of this equation leads to the logistic ODE:

      (3.13)StartFraction italic d n Over d tau EndFraction equals n left-parenthesis 1 minus n right-parenthesis

      where n is the population in units of carrying capacity left-parenthesis n equals upper P slash upper K right-parenthesis and tau measures time in units of 1/r.

      For systems, we can consider the predator-prey model in an environment consisting of foxes and rabbits:

      where:

r left-parenthesis t right-parenthesis = number of rabbits at time t
f left-parenthesis t right-parenthesis = number of foxes at time t
italic b r left-parenthesis t right-parenthesis = birth rate of rabbits
minus italic a r left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis = death rate of rabbits
b = unit birth rate of rabbits
minus italic p f left-parenthesis t right-parenthesis = death rate of foxes
q f left-parenthesis t right-parenthesis r left-parenthesis t right-parenthesis = birth rate of foxes
q = unit birth rate of foxes.

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