Past exam of the topics in applied mathematics course of the University of Oxford / 2010 / A / 1 / b / ii

The four nonnegative equilibria are

0, a_{1}, a_{2}, a_{3}

. Since all factors are positive below

a_{1}

, the sign of

h (N)

on consecutive intervals is

(0, a_{1}) : +, qq u a d (a_{1}, a_{2}) : -, qq u a d (a_{2}, a_{3}) : +, qq u a d (a_{3}, \infty) : - .

The phase line therefore points away from zero, towards

a_{1}

, away from

a_{2}

, and towards

a_{3}

. Equivalently

h^{'} (0) = a_{1} a_{2} a_{3} > 0

, and at each simple positive root

a_{i}

h^{'} (a_{i}) = - a_{i} \prod_{j \neq = i} (a_{j} - a_{i})

. Their signs alternate negative, positive, negative.

Hence extinction and $a_{2}$ are unstable, while $a_{1}$ and $a_{3}$ are asymptotically stable. Initial positive populations below

a_{2}

tend to

a_{1}

; those above

a_{2}

tend to

a_{3}

; an initial population exactly at

a_{2}

remains there. This is bistability with a threshold separating the two basins.

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