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)