Model of prey flux, D, with with C > 0 representing
ratio of intrinsic search to handling time

Kent, A., Doncaster, C. P. & Sluckin, T. Ecological Modelling (in press).

Set prey rate eqn to zero to get prey isocline:

Find apex of prey isocline:

Set predator rate eqn to zero to get predator isocline:

so: n* =

Elements of Jacobian

A
. dfs/ds at
s*,n*

At s*,n* :

simplify :

B
. dfs/dn at
s*,n*

At s*,n* :

C
. dfn/ds at
s*,n*

At s*,n* :

simplify :

D
. dfn/dn at
s*,n*

At s*,n* :

simplify :

Jacobian matrix is then given by :

J =

J =

J* =

trJ is the trace of Jstar, given by A
+D
; detJ is the determinant of J, given by AD-BC
.

Boundary between unstable and stable focus is given
by trJ=0 and detJ>0, at:

trJ = 0:

detJ = 0:

so must have D > -s

Boundary between stable focus and stable node is given
by (trJ)^2 - 4.detJ = 0, at

Metadiagram

Boundary between unstable and stable focus:

Boundary between stable focus and stable node (for v
= 0.1):

[D2 meets the condition for a stable node, that trJ
< 0]

[D3 meets the condition for an unstable node, that trJ
< 0]

This is the prey isocline for the full model:

prey rate = (D+s)(1-s) - sn/(C+s)